Jπ = 0+ states in the (fp) shell excited in the (3He, n) reaction

Nuclear Physics A243 (I 975) 269-297; Not to bc reproduccd

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J” =: 0+ STATES N EXCITED

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THE (fp) SHELL

Il’d THE (3He, n) REACTION

W. P. ALFORD t, R. A. LINDGREN tt, D. ELMORE and R. N. BOYD Nuclear Structure Research Laboratory, University of Rochester, Rochester, NY ttt Received 2 January 1975 Abstract: The (aHe, n) reaction has been used to study strongly excited 0” states in nuclei in the fp region. Measurements were made on targets of 46*48*5”Ti, 5o*52*54Cr, 54*56*ssFe and 58*60*62*64Ni at an energy of 15 MeV. In addition to the ground state transitions, L = 0 transitions to excited states were identified in all residual nuclei except 64*66Zn. Many of the observed states can be identified as analogues of low-lying states in isobaric nuclei, and can be predicted by the pairing vibration model (PVM). In many cases, low-lying excited states are seen which are not predicted by this model. Present results are consistent with the view that these states represent fragments of the expected PVM state. A comparison of present results with available (p, t) data leading to the same final nucleus suggests that little mixing occurs between states with different PVM configurations.

E

NUCLEAR REACTIONS 46.48. SOTi, 50.52~54Cr, 54.56.58Fe, 58.60.62.64Ni(3He, n) E = 15 MeV; measured ~((7, ,Q. 48~50~52Cr, 5% 54. 56Fe, 56.58.60Ni, 60.62.64.66~~ levels deduced J, n, L, 5’. DWBA analysis, model comparison. Enriched targets.

1.Iutroduction The importance of pairing forces in nuclei has been recognized for many years. One of their important consequences is the appearance of states of J” = O+ in which the microscopic structure of the state is described as a highly correlated superposition of zero-coupled pairs. The ground states of doubly even nuclei have this character and excited states with this structure may also arise. In many cases these excited states can be identified as analogues of ground or low-lying states of isobaric nuclei. The experimental identification of pairing states rests on the fact that the cross section for the L = 0 two-nucleon transfer reaction is enhanced by the pairing correlations ‘). The signature of these states is the appearance of a strong L = 0 transition in a two-nucleon transfer reaction connecting the state with the ground state of the target. By now, a great deal of experimental information is available on pairing states in the fp shell populated in (t, p), (3He, p) and (p, t) reactions, and many general features t Present address: Physics Department, University of Western Ontario, London, Ontario, Canada. tt Present address: Nuclear Science Division, Naval Research Laboratory, D.C. 20375. ttt Supported by a grant from the National Science Foundation. 269

Washington,

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of the data have been explained in terms of the pairing vibration model. “) Studies of the (3He, n) two-proton transfer reaction entail experimental difficulties and only a few measurements have been reported in the (fp) shell 3-7) until fairly recently. In the present work, L = 0 transitions have been observed in the (3He, n) reaction on targets of the even stable isotopes of Ti, Cr, Fe and Ni providing new information on pairing states in the vicinity of the ’ 6Ni closed shell. These measurements represent a continuation and extension of measurements of ground state transitions reported earlier “). Similar results, mainly at an incident energy of 18 MeV, have been reported recently by the Munich group ‘)_ Somewhat less extensive results have also been obtained by Fielding et al. ’ “) at 25 MeV. 2. Experimental A diagram of the beam pulsing system for the University of Rochester MP tandem is shown in fig. 1. The beam from the ion source is chopped at a frequency of 5 MHz, and the resultant beam bursts are velocity modulated in a conventional klystron buncher. After acceleration, the bunched beam is swept across an exit slit by the post-acceleration chopper. The phase of the post-acceleration chopper is adjusted to center the beam on the exit slit at the peak of the current pulse.This system results in a considerable improvement in the beam quality, both with respect to time resolution and dark current between pulses. Equally important, it provides a reference timing signal which is independent of ~uctuations in the transit time of the beam through the accelerator. In typical operation, average beam currents of 150 nA were available with an overall system timing resolution of 0.75 nsec. After the final chopping, the beam is focussed through a 0.48 cm diameter tantalum aperture mounted 1.3 cm from the target. The aperture plate is insulated and biased to repel secondary electrons from the target. After passing through the target, the beam is stopped in a 0.025 cm tantalum beam stop. The target and beam stop are connected electrically to the scattering chamber, which is insulated from the beam line and serves as a Faraday cup. The scattering chamber consists of a 10.2 cm diameter stainless steel tube with a wall thickness of 0.08 cm. Over the range of neutron energies of interest, neutrons from the target are attenuated by approximately 5 % by the wall of the scattering chamber ll). The targets of 46,48950Ti, 54956,58Fe and 58* 60* 62v ‘j4Ni were self-supporting rolled foils with thicknesses from 1 to 2 mg/cm’. The sop52*54Cr targets were prepared by heating isotopically enriched Cr,O, in a tantalum boat in vacuum. The oxide was reduced and the chromium target material evaporated on to gold backings about 200 pg/cm’ in thickness. These targets ranged from 150 to 500 pg/cm2 in thickness and showed a relatively large amount of carbon and oxygen as contaminants. Properties of the targets used in the measurements are shown in table 1. Since the measurements were undertaken to provide a comparison of cross sections

271

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TABLE 1 Properties of targets Target

Isotopic composition

Thickness

mass

oA

46 47 48 49 50

77.1 2.3 17.3 1.5 1.8

1.03

“sTi

48

99.4

0.93

5oTi

46 47 48 49 50

3.1 2.3 22.8 2.0 69.7

1.22

5OCr

50 52 53

96.8 3.0 0.2

0.16

50 52 53 54

4.3 83.8 9.6 2.4

0.55

50 52 53 54

0.2 7.0 2.2 90.6

0.22

46Ti

52Cr

54Cr

Target

(mg/cm)

Isotopic composition

Thickness (mg/cm)

mass

%

54Fe

54 56

98.6 1.4

1.25

56Fe

54 56 57 58

5.8 91.7 2.2 0.3

2.20

s8Fe

54 56 57 58

2.3 29.5 1.2 67.0

1.00

58Ni

58

100.0

1.41

60Ni

60

100.0

1.50

62Ni

58 60 62

0.5 0.8 98.7

0.45

64Ni

64

99.8

1.24

on different targets, considerable care was taken in measuring target thickness. Thicknesses were determined initially by observing the energy loss of a-particles passing through the foils. For foils of natural nickel and copper, for which the thickness could be measured by weighing, the method was found to give accurate results with an uncertainty of less than 5 %. Some of the target foils of separated isotopes were found to be quite non-uniform and to contain many small pinholes. For these targets the energy loss measurements were unreliable, and the average thickness of each target was eventually measured by observing the elastic scattering of low energy a-particles. This procedure was accurate and reproducible within counting statistics for the natural foils of known thickness. It indicated that some targets of separated isotopes had non-uniformities ranging up to f 30 %. A diagram of the detector system is shown in fig. 2. Three neutron counters, labelled 2,3,4, were mounted at angular separations of 5’ from one another on a cart which could be rotated about the target. A flight path of 4 m was used in the

273

t3He,n)

P -RECCIl ADC- 1

TOF ADC-3

11

11

COMPUTE2 ROUTING VIA ADC.1

Fig. 2. Detector system used in neutron time-of-Right measurements.

measurements. Neutrons were detected in 10.2 cm diameter NE 213 liquid scintillators of thickness 2.5 cm or 3.8 cm mounted on 12.7 cm photo-multipliers. The outputs of each counter were a linear dynode signal and a fast timing signal from the anode. The fast signal provided the start signal for the time of Bight measurement, and for a measurement of the rise-time of the linear signal. This latter information was used to provide neutron-gamma discrimination. Gamma-ray pulses could be almost completely rejected with negligible loss of neutron pulses. This was checked by comparison of counts in given peaks in the neutron and total (neutron plus gamma) spectra. The data acquisition program permitted each spectrum from a given counter to be gated by a selected portion of one of the other spectra from that counter. In practice, the time-of-flight spectrum was usually sated by the neutron pulses from the n-y discriminator, and the proton recoil spectrum was gated by one group in the TOF spectrum. The gated recoil spectrum displayed both spectrum end point, and discrimi-

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W. P. ALFORD et al.

nator cut-off for a neutron group of known energy, and thus provided a measurement of discriminator bias level needed to determine detector efficiency. The fourth counter was used for timing stabilization. It consisted of a 1 cm thick plastic scintillator mounted on a 56 AVP Amperex photomultiplier, with the scintihator about 0.5 m from the target. The position of the intense y-peak from this counter was determined every few minutes by the computer. If the peak showed any deviation from its initial position an appropriate timing correction was applied to the TOF spectra before they were stored in memory. Using this technique, the timing resolution could be maintained at 0.75 nsec. over the periods of several hours often required to accumulate a spectrum. The gains of the three neutron detectors were initially adjusted to be approximately equal using a radioactive source. The final adjustment was made by requiring equaI measured cross sections at a fixed angie for neutron groups from the 1zC(3He, n)140 reaction. In practice, gains could be easily equalized to within a few percent by this procedure. The discriminator bias level was measured as described earlier and set to correspond to a recoil proton energy of 4.6 MeV. Recently, absolute efficiencies have been measured ‘“) for NE213 liquid scintillator for neutron energies and bias levels in the range used in these measurements. Absolute efficiencies for the counters used in these measurements were obtained by interpolation of these published results. Neutron time-of-flight spectra were measured at angles of 0”, 5” and 10” for all targets at a beam energy of 15 MeV. In addition, angular distributions were measured at 5” intervals out to at least 40” for targets of 46Ti, 48Ti, “Fe and “Ni. A part of the data handling program converted measured flight times to excitation energies in the final nucleus. The time calibration of the TAC was measured using fixed delay lines of known length. The energy calibration of the system was then obtained by observing neutron groups of known energy from thin targets of “C. With the relatively thick targets used in most measurements the conversion from flight time to excitation energy required a knowledge of energy loss in the target. This was obtained by requiring that the program locate the ground state group at zero excitation energy, by adjusting the beam energy in the target, using the measured time calibration. The correction to the beam energy determined this way was consistent with that estimated from the known target thickness. Using this procedure, the uncertainty in excitation energy of well-defined peaks is estimated to be zt50 keV. 3. DWBA cakdations An appropriate review of the theory of two-nucleon transfer reactions has been given by Towner and Hardy 13), and the most important resuhs for present purposes are taken from that paper. A zero-range interaction is assumed in the cafculations. For the L = 0 transitions of interest here, a pair of particles with total spin S = 0 and isospin T = 1 are assumed to be transferred into some shell model state, specified

275

(3He.n) TABLE 2 Optical parameters used in calculations Particle

3He Neutron Bound state

VW = - Kf(r,

h. (fm) 175.4 f,(E) ‘)

k,

1.14 1.17 1.30

ad-iVJ(r,

0.71 0.75 0.65

19.9

fzW

r~, al)S4iWDa,

1.53 1.26

0.85 0.58

g cf(r, r,, a,))+

fb f,(E)

V,(r.)+

1.4 1.3 1.3

6.2

Vs... Q . I AZ2 i $

1.01

0.75

If+, rr.o.,

a,...)l.

f(r, rR, aR)= [l +exp(r--rRA+)/aa]-‘, h(E) = 56.3-0.32E-24(N--Z)/,4, h(E) = 0.22E-1.56, .f&) = 13-0.25E-12(N--Z)/A, f* = 9.13(N-2)/A. “) Adjusted to reproduce binding energy.

by (rr, Z,j). If the spins of both the target and residual nuclei are zero, and initial and final isospin are T and T’, then the differential cross section can be written do do = const x (TM1 - IIT’M’)2~~Yf(n,

Z,j)B’(n, Z,j, 0)12.

The quantity Y* is the spectroscopic amplitude for the reaction while B” contains the DWBA integrals. The summation goes over all shell model states into which the transferred particles may be placed. The expression exhibits the coherence between reaction amplitudes for different shell model states. Only in the case of a simple form factor involving a single shell model state can this expression be factored into a product of a spectroscopic factor times a DWBA cross section. DWBA calculations of the two-nucleon transfer cross sections were carried out using the code DWUCK with a zero-range interaction. Optical model parameters for the calculations were taken from Urone et al. 14) for 3He and Becchetti and Greenlees Is) for the neutrons. The parameters used are listed in table 2. Initial calculations with DWUCK assumed the transfer of a structureless cluster of mass two and charge two in the reaction. Later calculations were made with a modified version of DWUCK using the Bayman-Kallio method 1“) to calculate the form factor with the two transferred particles in the p+ or f+ orbit I’). Although the magnitude of the cross section was dependent on the details of the form factor, the angular distribution at forward angles was not, at least for L = 0 transitions. This is illustrated in fig. 3 which shows the result of calculations for different form factors, all normalized to the same cross section at 0”. For a form factor involving both fS and p+ orbits, the magnitude of the angular distribution depends on the relative contributions from the two orbits. However the shape is not sensitive to this mixture. For a form factor of the form a(4)’ +Jl (p*)’ the DWBA cross section as a function of a is shown in fig. 4.

-

a2

276

W. P. ALFORD

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10 20

30

40

50

60

70

%.m.(deg) Fig. 4. Comparison of shapes and magnitudes of L = 0 DWBA cross sections for form factor afg2_1-2/l -a2p+2 involving both fs and p+ pairs in the transfer.

Fig. 3. Comparison of shapes of L = 0 DWBA cross sections for diierent form factors. The three calculations are normalized to the same value at 0”. All calculations used the optical potentials from table 2.

I 0

t

2

t

G

4 Excitation

6 Energy

s

IO

(rev)

Fig. 5. Zero degree cross section as a function of excitation energy for different form factors. The different calculations are normalized to the same value for the ground state transition.

The dependence of the DWBA cross section on the excitation in the final nucleus (or reaction Q-value) is shown in fig. 5 for three different form factors, The calculations for different form factors were normalized to the value for (f$ for the transition of highest neutron energy. The Q-dependence is somewhat different for the different

277

We, n)

form factors, with the results for the cluster transfer giving a rough average of those for (fS)2 and (P+)~. Since the transitions of interest are expected to involve a mixture of both (fi)” and (p+)” components in the form factor, the cluster transfer calculations were used in the initial comparison between DWBA and experimental results. There have been recent indications that two-step processes can be important in two-particle transfer ‘**19) and (3He, t) reactions ‘O). To investigate this in the present case some calculations using the code CHUCK “‘) were also carried out assuming the (3He, n) reaction to be a two-step process proceeding through intermediate deuteron channels. While the magnitude of the calculated cross sections was different from that given by DWUCK, the relative cross sections at forward angles were not affected. Thus the spectroscopic results of this paper, which depend on these relative cross sections at forward angles, are based on the DWUCK calculations. The identification of O+ states rests on the characteristic decrease in cross section between 0” and 10”. Calculations for L = 1 and 2 were also carried out for different form factors, and it was found again that, while the magnitude of the cross section was sensitive to the details of the form factor, the angular dependence was fairly insensitive to these details. Typical results are shown in fig. 6 to illustrate the distinctive character of the L = 0 transitions. 4. Experimental results Neutron time of tlight spectra at 0” are shown in figs. 7a and 7b. Excitation energies in the final nucIeus are shown for groups which can be identified as leading to,O+ 102.

I

I

I

I

_

60

70

54Fe [3He,n)56Ni

0

IO

20

30 ec.m.

LO

50

(deg)

Fig. 6. Angular distributions predicted by DWBA calculations for L = 0, 1, 2. Calculations used a form factor ($1’ for L = 0, 2 and (fJ.g$ for L = 1 and optical parameters from table 2.

218

W. P. ALFORD

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states. The energy resolution in these spectra is about 600 keV for the ground state groups of highest neutron energy. For neutron groups at high excitation energies, the resolution was about 300 keV, with roughly equal contributions from target thickness and from the timing resolution. Groups arising from chemicaI or isotopic impurities are labelled by the symbol of the impurity. All spectra are characterized by the intense ground state group, and an evaporation-like spectrum. The latter becomes increasingly prominent for different targets as the reaction Q-value of the ground state transition increases. With the energy resolution available in these measurements, only neutron groups leading to the ground, and sometimes the first 2+ state, were clearly and unambiguously resolved. The characteristic forward peaking of the L = 0 angular distributions permitted the identification of strong L = 0

Fig. 7a. Time-of-flight spectra at 0” for Ti and Cr targets. Time calibration is 0.4 nsec per channel. Groups identified as L = 0 transitions are labelled by excitation energy in the final nucleus. Groups arising from target impurities are Iabelled with the impurity.

C3k n)

279

Fig. 7b. Time-of-flight spectra at 0” for Fe and Ni targets. Time calibration is 0.4 nscc per channel. Groups identified as L = 0 transitions are labelled by excitation energy in the tinal nucleus. Groups arising from target impurities are fabeiled with the impurity.

W. P. ALFORD

280

10 20 30

40 50 60 70

et uf.

IO 20 30 40 50 60

70

10 20

30 40 50 60

70

hn, (deg) Fig. 8. Angular distributions for selected L = 0 transitions. The solid curves are the result of DWBA calculations with optical potentials from table 2 and cluster form factor.

Fig. 9. Angular distributions for L = 2 transitions. The solid curves are the result of DWBA calcuiations with optical potentials from table 2 and cluster form factor.

transitions, even though the groups of interest often were not well resolved. Groups from “C and I60 imptirities are prominent in some spectra. These were easily identified, and generally caused little trouble. Only in “Fe did an impurity group clearly interfere with a known group from the target.

%ll.(~)

%il.(deg)

Fig. 10. Partial angular distributions for transitions identified as L 2 0 by the characteristic decrease in cross section between 0” and 10”. Except for ground state transitions, interference from background or other groups often obscured the groups of interest at larger angles. Solid curves are the result of DWBA calculations with cluster form factor.

%?l.(Ck$l)

E!?

N

282

W. P. ALFORD

er al.

Fig. 11. Angular distributions for transitions identified as unresolved groups with L = 0 and L = 2. The broken curves are the result of DWBA calculations with the separate L-values and the solid curves the summed fit to the data.

Measured angular ~stribut~o~s for several f, = 0 transitions are shown in fig. 8. AU show the forward peaking which characterizes such transitions. The solid curves are the result of DWBA calculations using a cluster form factor. For known L = 0 ground state transitions, the forward peak is very well fitted by the DWBA calculations using the average optical parameters noted. Agreement with the data was not very good at the second maximum in the angular dist~bution. This could have been improved by adjustment of the optical parameters, but for the purpose of extracting relative spectroscopic strengths, it was judged preferable to use the single average set of parameters for the analysis of ah the data, without at~mptin~ to force a fit to the details of jndividu~ angular distributions. In some cases, low-fying L = 2 transitions were resolved. Two L = 2 angular distributions along with DWBA calculations with a cluster form factor are shown in fig. 9 to indicate the agreement obtained. Cross sections at angles of O*, 5” and IO” are shown in fig. 10 for transitions which were identified as L = 0. The criterion for this identification, the characteristic

P Target nucleus

TABLE 3 = 0+ states excited in (3He, n) on Ti, Cr, Fe and Ni targets Final nucleus

(hz”)

g

(0”)

(mb/sr) 48Cr

“OCr

=Cr

52Fe

54Fe

56Fe

s6Ni

ssNi

6 ‘Ni 6 ezn

62Zn 64Zn 66Zn “) ‘) ‘) ‘)

0 5.48 8.80 9.54 11.2 0 4.00 5.84 11.5 b) 11.9 I 0 5.65 6.85 7.93 9.46 13.38 0 4.20 8.08 8.60 ‘) 0 4.32 5.39 6.51 8.56 10.75 0 5.30 6.55 8.15 9.24 0 3.96 5.11 6.61 7.91 9.96 0 3.58 10.61 0 3.39 0 6.63 7.41 Od) 5.39 0 0

1.25 ho.05 0.59f0.06 0.66iO.06 1.09 kO.05 0.6 ho.2 1.23 50.05 0.2660.05 0.7l=tO.O6 0.7 hO.2

0.76 1.15 1.23 1.22 1.13 0.58 0.87 1.oo 1.23

1.64 0.51 0.54 0.90 0.5 2.12 0.30 0.71 0.6

0.67f0.06 0.30f0.05 0.16iO.05 0.10f0.04 0.18f0.04 0.62_10.09 1.17&-0.08 0.43 io.05 1.08 kO.07 0.3 +0.1 0.78 *0.04 0.24*0.04 0.27 50.05 0.45 kO.04 0.31 iO.06 0.59 kO.05 0.59 *0.03 0.52fO.l 0.37kO.08 0.29 co.07 0.27;0.08 0.59f0.03 0.47*0.04 0.14&0.03 0.65 kO.03 1.52f0.05 1.21 kO.05 0.45 +0.03 0.86&0.03 0.44*0.05 0.34kO.03 0.66f0.04 1.42&0.04 040~0.05 0.65hO.05 0.98 ho.03 0.38&0.05 ‘) 0.68 &-IO.04 0.59f0.03

0.45 0.85 0.90 1.02 1.13 1.26 0.73 0.98 1.06 1.06 0.53 0.79 0.86 0.91 0.98 0.99 0.38 0.68 0.76 0.86 0.91 0.59 0.76 0.78 0.83 0.83 0.76 0.48 0.68 0.64 0.35 0.51 0.59 0.59 0.56 0.49 0.64 0.38 0.28

1.49 0.35 0.18 0.10 0.16 0.49 1.60 0.44 1.01 0.3 1.47 0.30 0.31 0.49 0.32 0.59 1.55 0.16 0.49 0.34 0.30 1.0 0.62 0.18 0.78 1.83 1.59 0.94 1.26 0.69 0.97 1.29 2.40 0.68 1.16 2.00 0.59 1.79 2.11

Cluster form factor, optical potentials from table 2. Broad group consistent with two levels having energies given. Obscured by I60 contaminant line. Energy given is that of total group. Group appears broad and may include contribution from l*C contamination

on target.

284

W. P. ALFORD

Ed al.

decrease in cross section between 0” and lo”, is seen to be satisfied in every case. The identification is perhaps questionable for the 6.85 MeV state in “Cr, though the data there are consistent with L = 0. It is probable that some of these groups are actually unresolved doublets or multiplets. Even if this is true the forward angle cross section can provide a reasonable estimate of L = 0 strength. This is indicated in fig. 11 where the transition to the 6.61 MeV state in 56Ni is seen to involve a superposition of L = 0 and L = 2 angular distributions, with the L = 0 component clearly dominant at angles forward of 10”. Table 3 lists the energy and zero degree cross section for the states which could be identified as J” = O+ states from these measurements. The uncertainty in excitation energy is estimated to be +50 keV for stronger transitions at low excitation. For some of the weaker states or states at high excitation with high background, the uncertainty is estimated to be fO.l MeV. The uncertainties quoted for the 0” cross sections arise from counting statistics and background subtraction only. An additional overah uncertainty of 10 % in absolute cross sections arises from the efficiency calibration of the counters. Finally, uncertainties in target thickness measurements contribute about 20 % uncertainty between different targets. The primary purpose of these measurements was to provide a comparison of intrinsic strengths for transitions to O+ states. As a first step, DWBA calculations with a fixed form factor can be used to account for the Z- and Q-dependence of the cross sections. In the absence of a detailed knowledge of the form factors involved, the results shown in figs. 3 and 5 indicate that the cluster form factor provides a reasonable approximation. This is the DWBA cross section listed in table 3 used to obtain the ratio dcrexp/d~nW.This ratio is taken as an estimate of intrinsic strength, to be compared with model structure calculations. The normalization of the DWBA calculations has been chosen to give a ratio of unity for the 54Fe + 56Ni (g.s.) transition, and the strengths shown in table 3 are alI taken relative to this value. Since the magnitude of the DWBA cross section depends strongly on the form factor, a determination of absolute intrinsic strength with a properly normalized DWBA calculation should permit a determination of the form factor itself. This problem will be discussed in a later paragraph. 5. Discussion Since the states observed in these measurements are those with strong pairing correlations it is natural to attempt to interpret the results in terms of the pairing vibration model 22*‘“). This model assumes the existence of a large number of states above and below an energy gap. Particles interact through a pairing force with a strength that is small relative to the magnitude of the gap. In the model, a pair of holes with angular momentum coupled to zero below the gap constitutes a removal quantum, and a zero coupled pair above the gap an addition quantum. Each state of interest is specified by the number of addition and removal quanta, N, and N,, their

We, 4

285

isospins T, and T,, and the resultant isospin IT,- TJ S T S Ta+ T,. In spite of its simplicity, the model is able to provide a convenient and useful framework within which the general features of the present results can be understood. The failures of the model can then provide some insight into the shell model structure of the states involved. In the (3He, n) reaction on a spin-zero target the cross section for addition of a zero coupled pair above or below the gap may be obtained from the results of ref. 23)_ In the first case an addition quantum is created, and

x

(Ta+w~+T*+3)gT 2T, -i-3

T *,+T ” a

@wa+2)6 =

ZT,-1

1+ (2)

T’*,T*-1

In the latter case a removal quantum is destroyed, and

These expressions focus on the isospin structure of the states involved. The coefficient U is a normalized Racah coefficient, the unprimed quantities refer to the initial target state and the primed quantities to the final state. The quantities Q, and Q, are essentially the DWBA cross sections and therefore account for the reaction kinematics.

6. Ground state transitions

Results of earher measurements of ground state strengths have been reported previously “)_ The present rest&s differ si~i~can~y from those for targets of 45748Til “Cr and “% due to the target non-uniformities discussed earlier. A comparison of measured relative strengths is shown in fig. 12 with the 54Fe + %i (g.s.) strength normalized to unity. In terms of the pairing vibration model (PVM), ground state transitions on the targets of Ti, Cr, and Fe involve the destruction of a removal quantum, and the predicted strength is given by eq. (3). Transitions on the Ni targets involve creation of an addition quantum, with the strengths given by eq. (2). The discrepancy between experiment and PVM predictions increases in moving away from the closed shell at ‘%Ji to lighter nuclei. This can be understood from the fact that the number of available states in the f* she@ is not large compared with the number of excitation quanta, as is assumed in the PVM. The large difference in experimental strengths between Fe and Ni targets arises from the difference in form factors for addition quanta (mainly (P~)~) and removal quanta (mainly (fS)2). This difference has been

286

W. P. ALFORD

3I’

A’

/

P

et al.

GROUJD STATE TRANSITIONS ---A-- PAIRVlMATC+4MODEL -O-

EXPERIMENT

A/’ A----A /

Fig. 12. Comparison between measured strengths S = da;,,fda nw and predictions of the pairing vibration model for ground state transitions. DWBA calculations employed a cluster form factor and were normalized to give agreement with the data for the $*Fe + a6Ni transition.

r-

----_-.

3

-.-

.-_I_.---.- ---

GROUND STATE TRANSITIONS --A---

SHELL MOD&

-o-

EXFER.~~NT

0

Fig. i3. Comparison between measured strengths and predictions of the simpIe shell model for ground state transitions. DWBA calculations employed (fg)’ form factor for Ti, Cr, and Fe targets and (p# for s8*60Ni targets. Calculations were normalized to give agreement with the data for the s4Fe + S6Ni transition .

t3He,n)

287

ignored in using the same cluster form factor in all DWBA calculations for the PVM analysis. Agreement for the nickel isotopes can be greatly improved if the calculations for these isotopes are normalized separately, which is equivalent to choosing a different form factor. A slightly improved mode1 is provided by a simple shell mode1 in which the ground states involve zero coupled pairs in the f4 shell for Ti, Cr and Fe target nuclei and pairs in the ps shell for the 58*60Ni targets. The predicted strengths in this case can be obtained from the results of Towner and Hardy ‘“). For the particular case of an L = 0 transition between Of states, the relative spectroscopic strengths are readily evaluated, using eqs. (4)-(11) of that reference. The comparison with experiment in this case is shown in fig. 13. The experimental strengths have been extracted using DWBA calculations with the optical parameters given in table 2, with form factor for (f;)2 transfer for Ti, Cr and Fe targets and (p+)’ transfer for ‘a* 60Ni targets. The mode1 is not applicable to 62164Ni targets since orbits lying above p+ must be included for the neutrons in the target ground state. Normalization of the DWBA was again chosen to give agreement with the strength of unity predicted for the s4Fe 3 56Ni (g.s.) transition. As can be seen in fig. 13, agreement between experiment and model predictions is still only fair for Ti and Cr targets, and for ‘**60Ni, experimental results are now low by a factor of nearly three. Although this is a large discrepancy, it can be readily understood in terms of excitations out of the f5 shell in the Ti, Cr and Fe ground states. In the simplest case of excitations from the f; to the pt shell, the form factor for the transferred pair can be written as C,(fJ2+CJpf)2 with C, x 1 and CI > C,. Since the DWBA cross section for the (P+)~ form factor is an order of magnitude larger than that for (f+)2, a relatively small (pa>2 amplitude in the form factor results in a large increase in the cross section given by eq. (1). This has been illustrated in fig. 4 for a form factor which reduces to (f;)’ or (P+)~ in the limits tl = I or 0 respectively. The dependence of the cross section on a is shown in fig. 14. It is seen that for c1% 1, the cross section is a sensitive function of a, but for 0.7 2 CI2 0, the dependence is relatively slight. If it is assumed that the transitions studied here involve a form factor a(fG)2+ then the model comparison may be used to estimate CIi.e. the relative Jl --arcs, contributions of (f3)2 and (pb)2 pairs in the transitions. The previous normalization of the DWBA using the ‘“Fe target presents difficulties however, because of the strong dependence on a for a x 1.An approximate normalization can be obtained by noting that for the Ni targets, the form factor is expected to be predominantly (p,)‘, corresponding to a NN0.The result shown in fig. 14 indicates that the assumption a = 0 would give an uncertainty of less than 20 % for the normalization obtained by comparing the experimental results for ‘*Ni with DWBA calculations. _DWBA calculations have been carried out for a form factor a(f~)2+J1 - G(jQ2 for the Ti, Cr and Fe targets. Using the normalization obtained for the ‘*Ni -+ 60Zn

288

W. P. ALFORD

et al.

Fig. 14. Variation of 0” cross section for L = 0 transitions involving (f$* and (p# The form factor is given as a function of the parameter cc.

contributions.

TABLE 4

Conf+guration mixing in two-particle Target

.=Ti

&*Ti

a

0.90

0.88

“) Form factor: a(f;)*t_v’l

0.94

form factors “)

500

s2Cr

54Cr

54Fe

56Fe

s8Fe

0.90

0.94

0.93

0.94

0.94

0.94

--(zz (p.#.

(g.s.) transition with CL= 0, these calculations have been compared with measured ground state cross sections to obtain the values of the parameter a shown in table 4. It is seen that small variations in the (P+)~ component in the form factor can account for the deviations between the measured strengths and predictions of the simple shell model. The above result is expressed in terms of the relative contributions from f%and pt pairs in the form factor. It would be of more interest for a model comparison to relate these results to the amount of excitation from the f%to the ps shelf in the nuclei studied, and an attempt was made to do this, using very simple model wave functions. Specifically, it was assumed that the ground state wave functions $(Ti, Cr, Fe) = ~(f&,(p&Z&+l/X

-j?2(f+)~%,z(p&“,~2,

(4)

with fi % 1. Except for J4Cr, 56*58Fe, m = 0. The seniority of each group is equal to zero, and the isospin equal to T, for that group. The second term of eq. 4 must include both neutron and proton excitations. Since the neutron excess is not very large for the nuclei of interest, it was assumed that the excitations of neutron and proton pairs were equaliy important.

t3He,n)

289

The corresponding model wave function for the ground states of 58*60Ni and 6oj 62Zn can be written $(Ni, Zn) = /3(p+)& f

41- /12(f.&~o(p,)_?~~ ‘,

(5)

withp = 1. With such wave functions, the spectroscopic amplitudes 9*(lf3) and 9”(2p,) can be obtained from the results of ref. 13) and the DWBA cross section given by eq. (1) can be calculated. For a wave function as given by eq. (5), the resultant cross section shows little dependence on B for ,!?a I. Hence the normalization of the DWBA cross section was again obtained by fitting the calculated cross section to the experimental rest& for the “Ni + 60Zn(g.s.) transition. To obtain an estimate of p in eq. (4) for the Ti, Cr and Fe targets it was necessary to assume a fixed value of /3 for both initial and final states in a given transition. This is justified by the previously mentioned observation that only small variations in the amount of (p+)” contributions to the form factor are observed. With this assumption, DWBA cross sections could be calculated as a function of the single parameter /.I. It was found however that in specific cases the cross-section enhancements predicted with these model wave func~ons were not great enough to fit the measured cross sections, and no estimate of /J could be obtained. This result presumabIy reffects the inadequacy of the simple mode1 wave functions that have been assumed. It is well known that ground state wave functions in this region require consideration of the complete (fp) space 24). Because of the coherent contributions to the sum in eq. (l), relatively small contributions from the pb and f+ orbits would generally enhance the predicted cross section, and this enhancement could be great enough to produce agreement between model calculations and the experimental results. A quantitative confimation of this suggestion would require a shell model calculation taking account of the full (fp) space however, and this is well beyond the capabilities of available computing facilities at the present time. 7. Double adogies In states invoIving several quanta in addition or removal states, the transferred quantum may be coupled to excite double analogue states. Such states differ from a lower-lying state in the same nucleus only in the isospin coupling of one of the groups of quanta. Since the higher state is the double analogue of the ground state or a low-lying state of an isobaric doubly even nucleus, the location can be reliably cafculated. In cases in which the state is expected at low excitation with reasonable strength, it has been observed. Since the form factor for the transition to the double analogue state is the same as that for the ground state of the same nucleus, a comparison between observed and predicted strengths should provide a particuIarIy good test of the model predictions.

W. P. ALFORD

290

et al.

TABLE 5 Double anatogue states in Ct, Fe, and Zn isotopes Final nucleus +Zr s2Fe “vh

6%

0 8.80 0 8.60 0 6.63 7.41 0 5.39

1.64 0.54 1.60 0.3fO.l 2.40 0.68 1.16 2.0 0.59

0.33

0.11

0.19

0.08

0.48

0.50

0.30

0.33

(4, ow, (4,2NO, (2, w, (2,2)(0, (0, om

ON 012 0)O w 0)O

(0,0)(2,2)2 <0,0)(3,1)1 (0,0)(3,3)3

Table 5 lists the states identified as double analogs in these measurements, along with the PVM description of each state. The strengths shown in table 5 are the same as those in table 3, and are obtained with the assumption of the same cluster form factor for each transition. The observed ratios of strengths are somewhat greater than model predictions for 48Cr and 52Fe, and very close to model predictions for the Zn isotopes. This probably indicates that the excited states in 48Cr and “Fe contain slightly more p-state excitations than the ground state, Ieading to an enhancement of the cross sections. This does not occur for 60*“Zn, since only states above the f+ shell are available to perturb the double analogue states, and small mixtures of these states would have relatively little effect on cross sections. The only ambiguity in this comparison arises for 6oZn. The state at 6.63 MeV is not predicted by the model, and its nature is unknown. If this were to be considered a fragment of the (N,, T,)(N,, TJT = (0,0)(2,2)2 state, then the experimental ratio of strengths would be slightly greater than the theoretical ratio, but agreement would still be good. 8. Addition-removal triplets For targets of 46‘48, SOTi, SO, 52~r and s4Fe, the ground state contains only removal quanta, at least in a first approximation. Excited state transitions on these targets are described in the PVM as coupling of a single addition quantum (T = 1, T, = - 1)with the removal quanta in the target (T = Ti,T, > 0)to form a triplet 1 are analogues of known with isospin Tf = T,, Ti* 1. The states with Tr = Ti,Ti-Istates in isobaric nuclei, and their location can be predicted. The states with Tf = Ti1 can only be excited in the (3He, n) (or possibly (p, t)) reaction. For targets of s4Cr, s6*58Fe the target already contains at least one addition quantum. This leads to the existence of an additional PVM state in the final nucleus, as well as the triplet. Table 6 lists those excited states observed in the present measurements, in addition to those already listed in table 5. Most of these may be correlated with the predicted

291

C3He. n) TABLB 6 Addition-removal Final nucleus

multiplets in Cr, Fe and Ni S(G)

(N,, G)(N.,

Analog state

T.)T

r=Sm

5.48 9.54 11.2

“) 4.20 8.08

1.76

1.50

0.33 0.50

0.30 0.71 1

0.60

0.6

0.33

2

11.97

‘OV(3.56)

0.07

3

17.11

5oTi(3.88)

3 4

13.53 19.87

SZV(2.39) szTi(g.s.)

1)O 1 2

8.05 11.17

52Mn(2.47) =Cr(2.66)

2 3

10.78 14.81

54Mn(2.12) 54Cr(g.s.)

3 4

14.1 20.7

56Mn(2.47) %r(g.s.)

(5, l)(l,

1)O 1

10.27

48V(3.701)

2

13.75

48Ti(4.97)

(4,2)(1,

I)1

0.17

‘) 5.65 6.85 1.93 9.46 13.38

thee

0.51 0.90 0.5

‘) 4.00 5.84 11.5 11.9 1

exp

0.35 0.18 0.10 0.16 i 0.49 0.04 0.44 1.01

‘) 4.32 5.39 6.51 8.56 10.75

0.30 0.31 0.49 0.32 I 0.59

‘) 5.30 6.55 8.15 9.24

0.76 0.49 0.34 1 0.30

:; 3.96 5.11 6.61 7.91 9.96

0.62 0.18 0.78 I 1.83 1.59

3.58

1.26

“) 10.61

0.69

‘) 3.39

1.29

0.6

0.71

0.55

(3,3)(1, 0.62

1)2

0.35

0.25 0.33 0.50 0.17

2.30

1.50

0.60

0.42

0.55

(3, l)(L

(2,2)(1.1)1

0.33 0.07 0.66

0.19

0.15d)

0.10 0.17 0.07 0.33

(2,2)(2,0)2 (2.2)(2,2)2

1.19

1.50

(1, l)(l, 1)O

0.50 0.17 0.83 0.02 0.10 0.08 1.0

0.55

0.12

1 2

7.83 9.98

YIo(l.45) 56Fe(g.s.)

2 3

10.56 14.49

58Co(l.81) 58Fe(g.s)

(1, 1)(2,0)1 (1, 1)(2,2)1

(1,1)(3,

I)2

‘) Calculated using cluster form factor for DWBA. ‘) Calculated with simple shell model assuming transfer pair in pi state. Identical with PVM resuhs except for 56Fe, 58Ni, 60Ni. “1 Not observed. “) G = T, state not observed. Ratio given for assumed (2,2)(2,2)2 and (2,2)(2,0)2 states.

292

W. P. ALFORD

et of.

members of the addition-removai triplets. The states with T, = Ti may be identified by their correspondence with the known analogs in isobaric nuclei which are also given in table 6. In every case except 56Fe and 60Ni the expected state is observed with a cross section in reasonable agreement with predictions. The predicted location of the state with Tr = Ti+ 1 is also given in table 6, but this state has been identified only in 56Ni. In general, this state is predicted to be at high excitation energy in a region of high background, and the predicted strength is small. In 48Cr 52Fe ‘ST6oNi only a single L = 0 transition is observed to a state below the known loweit Tr = Ti state. This is probably the expected state with Tf = Ti - I. In other cases, two or more low-lying L = 0 transitions are observed, suggesting a fragmentation of the predicted state. This suggestion is supported by a comparison of relative strengths for transitions to states with T, = Ti and Tf = Ti - 1. The experimental relative strengths listed in table 6 are the same as those in table 3 obtained using a cluster transfer model. The theoretical ratio of strengths, r = S(Ti)lS(Ti - 1) is calculated from results of ref. 13) assuming that the transferred pair is in the p+ shell. This is identical with the PVM result for targets which contain no addition quanta in their ground states. For “*Cr and “Fe, for which a single state is identified with Tf = Ti- I, the experimental ratio agrees with theory to within 15 % and 50 % respectively. The predicted ratio is based on the assumption of the same form factor for both states in each nucleus. Considering the deviations observed for the strengths of double analogues in these same nucIei, the agreement found here is satisfactory. In ‘*Ni, the state at 3.58 MeV may be identified with the predicted (N,, T,) (A$, T,) T = (1,1)(2,0) 1 state even though the measured strength ratio is in poor agreement with predictions. The (1,1)(2,2)2 state, seen at 10.61 MeV, is predicted to be weakly excited. The experimental cross section is much larger than expected; no explanation for this is apparent. In particular, there is no indication of experimental uncertainties large enough to explain the discrepancy, even though the state occurs at high excitation. For “Cr 52Cr 54Fe and 56Ni no single state with Tf = Ti - 1 contains the predicted stren’gth. &rch better agreement with the predicted strength ratio is obtained if all states below the lowest Tf = Ti state are assumed to be fragments of the predicted PVM state with Tf = Ti - 1. If the strengths of all these states are summed then in each case the strength ratio r is in quite reasonable agreement with predictions. In the case of 56Fe, the assignment of the (2,2)(2,2)2 configuration to the 9.24 MeV state and the (2,2)(2,0)2 configuration to the remaining states is based on the near equality between experimental and predicted strength ratios for the proposed assignment. The only observed states for which no PVM assignments are made are the state at 11.2 MeV in 48Cr, and the 11.5 MeV component of the broad state in 5oCr. In the latter case the state was broad but components could be clearly resolved, and the strength quoted is that for the whole group. Thus, in the analysis, the 11.5 MeV component has been treated as part of the expected (4,2)(1,1)2 state.

We, n)

293

9. Comparison with (p, t) results The (p, t) reaction has been studied between ground states of nuclei throughout the (fp) shell, and the results have been interpreted in terms of simple models 25). Some L = 0 transitions to excited states have been observed also, but there has been little attempt to correlate results in terms of the PVM, as is done here for the (3He, n) reaction. An important reason for this is that L = 0 cross sections to excited states are generally small. This is a reflection of the fact that most excited states involve the pickup off; particles, for which the DWBA cross section is small. In addition, at the high incident energies needed to study states up to high excitation, the experimental angular distributions do not show the prominent forward peak which makes identification of L = 0 transitions relatively easy as in the (3He, n) results. In cases in which the (p, t) and (3He, n) reactions can populate states in the same final nucleus, a comparison of cross sections in the two reactions may provide further information on the validity of the proposed model assignments for excited states. In many cases, the PVM would predict that a level populated in one reaction would involve a forbidden transition in the other. The occurrence of the supposedly forbidden transition may then be an indication of mixing of PVM states, or of an incorrect model assignment. Some available data for such a comparison are shown in table 7. For each reaction, the state of highest excitation listed indicates the energy range studied. The (p, t) data are taken from a number of different experiments at different incident proton energies. In most cases, a measure of the cross section is taken as the cross section at the peak of the second maximum, which occurs at angles between 20” and 40” depending on incident energy and reaction Q-value. Ground state cross sections are normalized to unity, and excited state cross sections are given relative to the ground state. For targets of 60Ni and 62Ni the cross sections reported are integrated cross sections, from 15” to 65” for 60Ni an; 12 5” to 60” for 62Ni For ’ 6Ni, Oi states of the PVM configuration (N,, T,)(N,, T,) 1 (l,l)(l,l) should be populated in both reactions. The states observed in (3He, n) are also populated in (p, t) up to the limit of the measurements. Note that the relative cross sections to the 3.96,5.11 and 6.61 MeV levels suggested here as components as the (l,l)(l,l) T = 0 state, are very different in the two reactions. Such differences presumably reflect the detailed structure of the states involved. On the basis of the PVM assignments given in table 6, some (3He, n) transitions would lead to states which could not be excited in a (p, t) transition. Examples are the states at 4.00 MeV in 5oCr, 8.08 MeV in 52Fe and 3.39 MeV in ‘joNi, which are seen in present results but not reported in (p, t) measurements. Transitions to the states at 4.20 MeV in 52Fe and 3.58 MeV in ‘*Ni should also be forbidden in (p, t). The (p, t) transitions are reported, but are quite weak. Similarly, the (p, t) transition is allowed on the PVM to levels at 2.65 MeV in 52Cr, 2.56 MeV in 54Fe, and 3.50 MeV in 56Fe, while the corresponding (3He, n) transition is forbidden. In these

1%‘.P. ALFORD

294

et af.

TABLE 7 Comparison

of (3He, n) and (p, t) measurements L = 0 transitions

We, n) Target PVM configuration

48Ti (4,2)(O,O)T

(P. t) S

(&)

0

5.84 I 11.5 11.9 I ‘OTi (3, 3)(O,O)T = 3

2.12 0.30

0.71 I 0.6

1.49 0.35 0.18 0.10 0.16 I 0.49

0 4.20

1.60 0.44

1

0

0.07 0.07

4.35 4.15

1.0 0.1

0 2.65

(2,2)(1,

1 0.03

(1, l)(O, 0)l

Y Y

0 4.14 5.31 6.84

Y

8.52

5zCr

+%r

(2,2)(0,0)2 (3, l)(l, 1)2 (3,3)(1, I)2

(3,3)(1,

54Fc

(2,0)(0,0)0 (3, l)(l, 1)O

(3, l)(l, 1)l (2,2)(0,0)2

0

1.47

(1, l)(O, 0)1 (ZO)(l, 1)l

1 0.17

0 2.56

4.32 5.39

0.30‘ 0.31 (2,2)(1,

1)l

0.03 0.02 0.02 0.06

6.316 6.406 6.507 6.102

(2,2)(1,

1)2 1 0.12 0.06 0.04

0 3.50 3.97 4.62

8.56 I 10.75

0.32, 0.59

0

1.55

5.30 6.55 8.15 I 9.24

0.76 0.49 0.34 I 0.30

(1, l)(l. 1)2 (2,0)(2,2)2

(2,2)(2,0)2 (2,2)(2,2)2

(1, l)(l,

d, 1)2

seFe

56Fe

WZr

‘)

s6Fe

5“Fe

0.49

“) 1)3

I)3

1.Ol 0.3

6.51

“)

(2,2)(0,0)2

(4,2)(1,1)2

8.08 8.60 Wr

I)3

(k’)

52Fe

50Cr

(2,2)(1,

(3. l)(O, 0)l (4,2)(1, 1)l

Target PVM configuration

do,,,

52Cr 0 5.65 6.85 7.93 9.46 i 13.38

(2,2)(0,0)2

da

SOCr = 2

4.00

(3, I)(% 0)1

Final nucleus PVM configuration

(1, 1)(2,2)3

*)

We, n) TABLE

295

7 (continued)

t3He,n) Target PVM configuration s4Fe (1, l)(O,O)l

s6Fe (I, l)(l, 1)2

(P, t) s

(&)

Final nucleus PVM configuration s6Ni t0,om 0)O

0 3.96 5.11 6.61 I 7.91 9.96

1.0 0.62 0.18 0.78 I 1.83 1.59

(1, l)(l, 111 (1, l)U, 112

0

0.94

s8Ni (O,O)(l, 1)1

3.58

1.26

10.61

0.69

(1, l)U,l)O

(1, 1)(2,0)1 (1,1)(2,2)1(?) (1,1)(2,2)2 (~1)(2,2)3

(I, w,

s8Fe 2)3

0

0.97

60Ni (0,0)(2,2)2

3.39

1.29

(1, 1)(3,1)2 (1,1)(3,3)3

“) Ref. 26). “) Ref. 27) ‘) Refs. zs: 29).

“) Refs. 2g. 30). “) Ref. 30). ‘) Ref. ‘I).

da -dw.,

(&)

1 0.001 0.004 0.008 0.015

0 3.95 5.00 6.69 7.96

1 0.01 0.007 0.04

0 2.94 3.50 5.96

0.05

14.47

1 0.02

0 2.28

0.04 0.04

4.34 13.76

Target PVM configuration 58Ni ‘) (0, w, 1)1

60Ni (0,0)(2,2)2

*)

62Ni (0.0)(3,3)3

‘)

‘) Ref. 32). “) Level reported, cross section not given.

cases, no (3He, n) transition was observed, but a transition with a strength could be obscured by other levels or by target impurity groups. The fact that transitions, which would be forbidden by selection rules based PVM, are weak or unobservable, indicates that there is little mixing between with different PVM configurations. This behavior is interesting in the light fragmentation of some PVM states noted earlier.

5 0.1 on the states of the

10.Summary and conclusions The present study has provided measurements of strong L = 0 transitions in the (3He, n) reaction on the thirteen stable doubly even isotopes of Ti, Cr, Fe and Ni. In addition to the ground state transitions, thirty-six transitions to excited states were observed. The general trends of the ground state transition strengths, and the nature of almost all excited states could be understood within the framework of the simple pairing vibration model with isospin. The observed deviations from predictions of this model

296

W. P. ALFORD

et al.

could be explained as arising from known limitations of the simple model. A quantitative treatment of the discrepancies was attempted, using a simple shell model, but it was concluded that the model was not adequate. An obvious extension of the model that is required is to use a larger vector space, but a truly realistic space would involve impossibly large shell model calculations. The observations for transitions to excited states in “Cr “Cr, 54Fe, 56Fe and 56Ni can be interpreted as a fragmentation of the strength of a single PVM state over severa levels, with no loss of predicted total strength. This indicates that the correlations in the wave function giving rise to the enhanced L = 0 transitions are not destroyed by the fragmen~tion of the state. Finally, a comparison of the present results with other available data on (p, t) reactions indicates that selection rules expected on the basis of the PVM apparentIy have an important influence on the cross sections. There appears to be little mixing of states with different PVM assignments, in spite of the extensive fragmentation indicated for some of these states. We are very much indebted to the operations staff of NSRL for their efforts in providing the pulsed beam for these measurements. SpeciaI thanks are due to Mr. T. Lund and Dr. K. H. Purser in this. We are grateful to Prof. A. Bernstein of MIT for the loan of the Ti foils used in the measurements. References I) S. Yoshida, Nuci. Phys. 33 (1962) 685 2) 3) 4) 5) 6) 7) 8) 9) 10) it) 12) 13) 14) IS) 16) 17) 18) 19) 20) 21)

0. Nathan, Symposium on nuclear structure, Dubna (IAEA, Vienna, 1968) p. 191 H. C. Bryant, J. G. Beery, E. R. Flynn and W. T. Leland, Nucl. Phys. 53 (1964) 97 R. G. Miller and R. W. Kavanagh, Nucl. Phys. A94 (1967) 261 D. D. Brill, A. D. Vonigai, V. S. Romanov and A. R. Faiziev, Yad. Fiz. 12 (1970) 253 [transl.: Sov. J. Nucl. Phys. 12 (1971) 1351 M. B. Greenfield, C. R. Bingham, E. Newman and M. J. Saltmarsh, Phys. Rev. C6 (1972) 1756 R. P. J. Winsborow and B. E. F. Macefield, Nucl. Phys. A182 (1972) 481 W. P. Alford, R. A. Lindgren and D. Elmore, Phys. Lett. 42B (1972) 60 D. Evers, W. Assmann, K. Rudolph, S. J. Skorka and P. Sperr, Nucl. Phys. A198 (1972) 268; A230 (1974) 109 H. W. Fielding, P. G. Brabeck, R. E. Anderson, D. A. Lind and C. D. Zaiiratos, Bull. Am. Phys. Sot. 19 (1974) 545 D. J. Hughes, B. A. ~~agurno and hi. K. Brussell, Brookhaven National Laboratory Report BNL-325, 2nd ed. (1960) M. Drosg, Nucl. Instr. 105 (1972) 573 I. S. Towner and J. C. Hardy, Adv. in Phys. 18 (1969) 401 P. P. Urone, P. W. Put, H. H. Chang and B. W. Ridley, Nucl. Phys. Al63 (1971) 22.5 F. D. Becchetti and G. W. Greenlees, Phys. Rev. 182 (1969) 1190 B. F. Bayman and A. Kallio, Phys. Rev. 156 (1967) 1121 P. D. Kunz, code DWUCK, modified by D. C. Shreve, Nuclear Structure Research Laboratory, University of Rochester W. W. Daehnick and R. Sherr, Phys. Rev. C7 (1973) 150 P. D. Kunz, private communication W. R. Coker, ‘I’. Udagawa and H. H. Woher, Phys. Rev. c7 (1973) 1154 P. D. Kunz, CHUCK, a coupled-channets two-step distorted wave computer code, University of Colorado, unpublished (1973)

We, n) 22) 23) 24) 25) 26) 27) 28) 29)

297

A. Bohr, Symposium on nuclear structure, Dubna (IAEA, Vienna, 1968) p. 179 B. F. Bayman, D. R. Bes and R. A. Broglia, Phys. Rev. Lett. 23 (1969) 1299 S. S. M. Wong and W. G. Davies, Phys. Lett. ZSB (1968) 77 B. F. Bayman and N. M. Hintz, Phys. Rev. 172 (1968) 1113 H. W. Baer, J. J. Kraushaar, J. R. Shephard and B. W. Ridley, Phys. Lett. 35B (1971) 395 C. A. Whitten, Phys. Rev. 156 (1967) 1228 J. B. Viano, Y. DuPont and J. Menet, Phys. Lett. 34B (1971) 397 T. Suehiro, Y. Ishizaki, H. Ogata, J. Kokame, Y. Saji, A. Stricker, Y. Sugiyama and I. Nonaka, Phys. Lett. 33B (1970) 468 30) H. Nann, W. Benenson and W. P. Alford, unpublished results (1974) 31) H. Nann and W. Benenson, Phys. Rev. 1OC (1974) 1880 32) D. H. Kong-a-Siou, A. J. Cole, A. Giorni and J. P. Longequeue, Nucl. Phys. A221 (1974) 45