High resolution inelastic proton scattering near A = 50

ANNALS

OF

PHYSICS:

High

51,

124-161 (1969)

Resolution

inelastic

Proton

Scattering

Near A = 50”

G. C. KYKER, JR.~, E. G. BILPUCH, AND H. W. NEWSON Duke University, Durham, North Carolina 27706

The yield of the deexcitation gamma ray following inelastic proton scattering to the first excited states of 4BTi, @Ti, “0, 64Fe, and &“Fe has been studied for proton energies between 2.5 and 3 MeV with energy resolution of approximately 1700 eV. For all targets except 64Fe, angular distributions of the gamma-ray yield have heen measured as a means of assigning spins and parities to resonances observed in the excitation functions. A total of approximately 160 resonances have been observed and assigned in this way. This work has shown the first excited state of 60Cr to have spin 2. In addition, the spacings of 5/2+ levels in the compound nuclei a7V, 61Mn, and 57Co are consistent with the dependences on neutron excess and shell structure observed in neutron total cross section measurements. There are also indications in the data of a maximum in the D-wave proton strength function in the neighborhood of mass number 47.

I. INTRODUCTION

Historically, the single most productive tool in the study of virtual states of nuclei has been the measurement of the neutron total cross section. This technique has the advantage of easy isolation of a single partial wave by the characteristic pattern of interference between an s-wave resonance and the local non-resonant cross section. A great deal of data of this kind now exists and over the last several years considerable progress has been made in understanding the average properties of neutron resonance levels: average level spacing, average strength, and so forth. In particular Newson and others (l)-(3) have been able to explain the pattern of average level spacings near mass number 50 very convincingly in terms of a dependence on compound nucleus excitation energy, neutron excess, and the effect of target nucleus shell structure. We have attempted in the work reported here to conduct a more or less parallel study of resonance systematics as observed in the inelastic scattering of protons by nuclei in the neighborhood of mass number 50. We have studied this process on targets of 46Ti, 48Ti, 5oCr, 54Fe, and 56Fe; of these, 4sTi was found to have too great * Work supported in part by the U.S. Atomic Energy Commission. t Based on a dissertation submitted in partial fuhillment of the requirements for the doctoral degree from Duke University. Present address: Oakland University, Rochester, Michigan.

124

INELASTIC

PROTON

125

SCATTERING

a level density to be systematically studied, while in the case of 54Fe only a few weak resonances were observed in the energy region accessible to us. In each of the remaining three nuclei a limited energy range could be studied intensively, and in each about 50 resonances were observed and assigned. Spin and parity assignments were derived from the angular distribution of the deexcitation gamma-ray yield at each observed resonance. These measurements alone do not yield an unambiguous assignment for each individual resonance. However, by taking into account the favoring of a few spin states by barrier penetrabilities, we can make firm enough assignments, on the average, to be able to draw conclusions about mean resonance parameters.

II.

EXPERIMENTAL

PROCEDURE

All the data reported here were taken utilizing the proton beam from the Duke University 3 million volt Van de Graaff accelerator. The general floor plan of the laboratory for this work is shown in Fig. 1. The Wisconsin type 1 meter cylindrical electrostatic analyzer described by Toller (4) was used for energy measurement. The beam is passed through the analyzer immediately after the energy control slit, which also forms the optical object for the analyzer. The resolution of the analyzer is

VAN

DE GRAAFF ANALYZING MAGNET CYLINDl$ICAL ELECTROSTATIC ANALYZER

TERMINAL+--, VOLTAGE c~,~y#.~L~ 4 -----

LABORATORY

FLOOR

0 -- -_:

Lzi!i9 TARQET

PLAN

DETECTOR FIG.

1.

Floor

plan of the laboratory.

ON TABLE

126

KYKER,

BILPUCH,

AND

NEWSON

controlled by the widths of this slit and the image slit located at the output end. For the current work the analyzer slits were adjusted for an energy resolution of approximately one part in 4000, or approximately the energy spread of the beam after the control slit (5). The voltage on the analyzer plates is supplied by a high-voltage rectifier circuit controlled, through a PetreeHenkel (6) circuit, by the setting on a precision potentiometer. Calibration of the analyzer for the current work was obtained by measuring a 0” threshold curve for the ‘Li(p, n) ‘Be reaction. We estimate our absolute energy calibration to be accurate within about 2 keV, while relative energies between resonances in a given target are reliable to about 1 keV. The targets were evaporated metallic films of separated isotopes prepared by Oak Ridge National Laboratory. These were deposited on Ta end caps, 2.0 in. in diameter and 0.01 in. thick, which formed the vacuum seal at the end of the target chamber. In each case the target layer was of the order of 25 pg/cm2, or between 1.5 and 2.0 keV thick to 3 MeV protons. The y-ray detector consisted of a 1.75- x 2-in. NaI(TI) scintillator coupled to an RCA-6342 photomultiplier. The resolution of this detector for y-rays in the range 780 to 990 keV was about 9 %. The detector was mounted within a 6-in.diameter lead shield to eliminate y rays originating from sources other than the target. The detector was mounted so that it could pivot around the target, but in such a way that the detector’s center of rotation was rigidly coupled to the target chamber. At its closest approach to the target the crystal subtended an effective solid angle of 0.063 sr, and the correction for finite angular resolution was insignificant on all but the most strongly anisotropic angular distributions. Conventional commercial circuits were used in analyzing and counting the y rays detected by the NaI crystal. Figure 2 shows a portion of the y spectrum (about 0.2-1.6 MeV) seen by the detector on and off the 2.689-MeV resonance in 5oCr(p, p’). One notable feature of the spectrum is the essentially flat region at energies above that of the 50Cr(P, p’) photopeak. (The small peak appearing near 1.04 MeV in this spectrum was not a consistent part of the 50Cr data and has not been identified). This we ascribe to higher-energy y-rays from proton capture in deposited carbon and perhaps in the target or other nearby materials. In order that the angular distribution data be meaningful, especially for weak resonances, it is necessary to subtract this contribution from the primary line being counted. For this purpose the amplified spectrum of pulses from the detector was coupled in parallel to two differential pulse height selectors which were set to pass the portions of the spectrum labeled A and B. The A gate is centered on the full-energy peak and is of such a width that about 80-85 % of the peak is counted. Counts from the two pulse-height selectors were registered in two printing scalars and their difference taken as the net yield of the first excited state y-ray. While this setting sacrifices much of the available yield it has the advantage of

INELASTIC

PROTON

127

SCATTERING

Y RAY

SPECTRUM CrJO t p -2.689 MN --- 2.662 MN

To bucking

0

10

20

3”

CHANNEL

.^

4”

,-. .^ 3”

-.

bd

NUMBER

FIG. 2. Gamma ray spectra from Tr +p. The full curve is the spectrum at a strong resonance in Tr(p, p’y); the dashed curve is the spectrum off resonance.

the energy region in which lower-energy y-rays (for example, the 0.50MeV line in Fig. 2) can affect the data. Another advantage of this setting is its relative insensitivity to baseline or gate width instability in the A-channel pulse height selector. While in Fig. 2 the two gates are indicated as of equal width, closer study of the spectra showed that the appropriate subtraction was achieved by running the B window about 10 % wider than the A window. The data on 46Ti, 4sTi and 56Fe were taken in this way and the 50Cr data, which were taken with gates df equal width, were corrected to be consistent with the others. 46Ti and 54Fe presented rather specialized background subtraction problems which will be discussed below. For all the target nuclei studied except 54Fe, the excitation energy of the second excited state is considerably more than twice that of the first, so that any competing inelastic scattering to the second excited state which decays by cascade through the first will produce a higher-energy ray than that from the first excited state. In this way the B window serves as a monitor on any (p, pZ) competition, whether the decay is direct or by cascade through the first-excited state. Nowhere in the present data was any indication of (p, pJ competition seen and indeed at these energies no significant amount is to be expected.

minimizing

128

KYKER,

BILPUCH,

AND

NEWSON

When the data were taken, significant absorption of y rays in the target backing was observed, particularly between 80” and 90”. In order to correct the angular distribution data for the resulting loss of counts, we measured the absorption of y rays from a radioactive source (60Co or 13’Cs) deposited on an identical backing and substituted for the target. The source duplicated as closely as possible the size and position of the beam spot on the target. Measured angular distribution data were corrected near 90” using the absorption numbers arrived at in this manner. For all the targets studied an excitation curve was measured at a detector angle of 130” to locate resonances and in general to define an energy region suitable for further study. Typically this region was chosen between an energy below which no strong resonances were observed and an energy above which a significant number of resonances were insufficiently well isolated for individual analysis. When the resonances in a given target had been located, angular distributions of the y-ray yield were measured on each one. These distributions were measured in 10” steps from 0” to 140” from the beam direction, except for about half the 5oCr + p distributions which were measured only between 0” and 80”. On each target a few angular distributions were also measured between resonances for background subtraction purposes. For all the excitation function data and for the angular distribution data on 56Fe and 5oCr, counts were taken against integrated beam. Because of an apparent instability in yield observed at times in this data, a different method was used for the angular distribution measurements on the Ti targets. In these cases a second, fixed scintillation counter was used as a monitor, and the number of counts in the movable counter taken for a given number of monitor counts. In fact, no great TABLE TARGET

I

CHARACTERISTICS

First-ExcitedState Target Isotope

Enrichment (%I

46Ti 48Ti Wr 64Fe ssFe

86 99 99 97 92”

Thickness= (h&m”) 25 27 33 27 24

WV)

Spin and parity

0.889

2+

0.991 0.780 1.39 0.845

(2+)

Energy

(& 2+

EnergyRange Studied WV) 2X-2.90 2.65-2.90 2.15-2.85 3.10-3.35 2.60-2.95

4 Except for 5eFe, thicknesses are as stated by the supplier. The thickness given for 6BFeis estimated from apparent resonance widths. * Natural iron target.

129

INELASTIC PROTON SCATTERING

difference in the quality of the data was observed between the two methods, and we concluded that our occasional stability problems arose from instability in the beam energy and counter electronics jitter.

III. EXCITATION

FUNCTIONS

Table I gives information on the targets studied. In each case the energy range indicated in Table 1 includes all the resonances to which we have attempted to assign spin and parity by angular distribution measurements. The 2+ assignments for the first excited states of 50Cr and 54Fe are bracketed in the table because at 130’ RELATIVE YIELD Ti’” ( p.p, )Ti46+ 0.89 MeV 7

l.O-

y

(I

.5 12

O-

34

5

6

I

i 150

2.55

5.0+

! o--

;..__,_.;_,

4, .3 44*I

I -

tr i O2 70

2.75

2.80

2.85

E, in MeV FIG. 3. Excitation function for 46Ti(p, p’) to the 0.89-MeV state.

2.90

y RAY SPECTRUM Ti46 t p E,= 2.664MeV

690 Ti”(p,p,)Ti*”

keV

990

IO

I 20

I 30

40

CHANNEL

50

keV

60

I 70

60

NUMBER

FIG. 4. Gamma-ray spectrum from @Ti + p at the energy of a very strong resonance in 4BTi(p, p’). The 46Ti target contains about 10 Y04HTi.

130° RELATIVE YIELD Ti4* (p,p,)Ti4’+0.99 MeVy

E, FIG.

in MeV

5. Excitation function for 48Ti(p, p’) to the 0.!-?9-MeV state.

INELASTIC

PROTON

131

SCATTERING

present these assignments rest only on analogy to other even-even nuclei. Our independent evidence for this assignment in the case of 50Cr will be discussed below. Figure 3 shows the excitation function for the 46Ti( p, p’) reaction to the 890-keV first-excited state of 46Ti, over the energy range 2.5 to 2.9 MeV. While data were actually taken over a somewhat wider range, Fig. 3 includes all the resonances on which gamma ray angular distributions were measured. For about 100 keV below 2.5 MeV, no resonances were observed with a “relative yield” greater than 0.15, and shortly above 2.9 MeV, a case or two was seen of strong resonances too closely spaced to be resolved. (The ordinates of Fig. 3 and of the other excitation function plots shown have themselves no significance; in Fig. 3, a “relative yield” of 1.0 corresponds to roughly 3 x 1O-g reactions per incident proton at the target). The 46Ti target contained 86.4 % of 46Ti and about 10 % of 4*Ti. Because the first excited state gamma rays of the two are of similar energy, this impurity occasionally gave rise to a background problem. Figure 4 shows an extreme example. This is a gamma-ray spectrum taken from 46Ti + p at a bombarding energy of 2.864 MeV, the energy of the strongest resonance observed in 48Ti(p, pJ. At this energy the 46Ti(p, pl) yield is not high, so that the effect is exaggerated; still the

130’ RELATIVE

i

YIELD

Cr5’ (p,p,) Cr5’ + 0.78 MeV Y

f3 w F s 0. s-1 k!

2.15

2.20

2.25

2.30

2.35

2.40 Ep in MeV

2.45

2.50

3.0

2.30

FIG. 6. Excitation function below 2.5 MeV for Wr(p,p’)

to the 0.78-MeV state.

132

KYKER,

BILPUCH,

AND

130” RELATIVE Cr5’(p,lj,)

9 w s Y 5 g

NEWSON

YIELD

Cr5’ + 0.78 MeV 7

1.0

2.50

2.55

2.60

2.65

2.00

2.85

2.70

4.0-

3.0 -

2.0-

1.0 -

2.70

2.75

E,, in MeV FIG.

I.

Tr(p,

p’)

above

2.5 MeV.

4sTi yield obviously will effect to some degree what we observe for 46Ti. With the gates set as shown in Fig. 4 about 5 % of the 4BTi yield will appear superimposed on the 46Tidata. We believe someof the very weak resonancesin Fig. 3 (for example, those at 2.745, 2.764, and 2.806 MeV) to be due to the 48Ti impurity in the target. Once the problem is recognized, about the only way in which it could distort the 46Ti data would be the occurrence at the same energy of a weak resonance in 46Ti(p, pl) and a strong one in 48Ti(p, pl); this would make the angular distribution from the weak resonance difficult to measure. However, it does not appear from the excitation functions that any such coincidences occur. Figure 5 shows the yield of the 48Ti(p, p’) reaction to the 991-keV state. The proton energy range shown is approximately 2.65 to 2.85 MeV. Because of the complicated resonance structure observed with this target, it was not felt to be practicable to measure angular distributions systematically on all observed resonances.The strongest resonances(those numbered in Fig. 5) have been studied in the limited energy range shown.

INELASTIC PROTON SCATTERING

133

The relative yield of the 50Cr(p, p’) reaction to the 780-keV state is shown in Fig. 6 and 7. Below 2.3 MeV there are no strong resonances.In order to increase the number of resonancesanalyzed the data was extended down to 2.15 MeV. The higher first-excited state energy (1.39 MeV) of 54Fe made it necessary to look for resonancesat a somewhat higher bombarding energy, and our maximum energy of about 3.3 MeV limited the region which we could study. The yield of 5’Fe(p, ply) near 3.2 MeV is shown in Fig. 8. When thesemeasurementswere made it was found that an accidental silicon contamination of the target backing gave rise to a high background of 1.77-MeV gamma rays from the YSi( p, pry) reaction. This reaction has strong resonances(7) near 3.1 and 3.3 MeV; in theseregions such a large background subtraction from the 54Fe data was necessary that angular distribution measurementswere not feasible. The yield of the 56Fe(‘. p’) reaction to the 845keV state is shown in Fig. 9. Here we have been able to measure angular distributions on all resonances of appreciable strength, i.e., on more than 80 % of the resonancesindicated in Fig. 9. Again we have limited angular distribution measurementsto a small energy region

130° 04

RELATIVE

YIELD

Fe54(p.p,)‘Fe54tl.39

MeVy

E,

in MeV

FIG. 8. Excitation function for 54Fe(p, p’) to the 1.39-MeV state.

134

KYKER, BILPUCH, AND NEWSON

0.

130’ RELATIVE YIELD Fes6 ( p.p, 1 Fe%+ 0.84 MeVy

0.

E,

in

MeV

FIG. 9. Excitation functionfor 6dFe(p, p’) to the 0.84-MeVstate. in which most resonances are reasonably strong and fairly well separated from one another. The shapesof the resonancesobserved in the various excitation functions shown here display some remarkably consistent features. With a few exceptions, the apparent width of the resonancesin a given reaction is quite uniform so that in most casesthe natural resonance width must be much smaller than the approximately 2-keV energy spread due to target thickness. Furthermore, these resonanceshave an apparently uniform shape. In all caseswhere the shape is well defined by the data we observe a steep rise (in a range of a few hundred eV) on the low-energy side; followed by a markedly slower faI1. This consistent asymmetry is apparently due to straggling of the proton beam in the target. Using the result of Symon (8)

INELASTIC

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SCATTERING

135

[which, for very thin targets, reduces to the Landau (9) straggling distribution] and known values for the beam spread and target thickness, we were able to reproduce the observed resonance shapes very satisfactorily. A similar effect was noted by Parks et al. (IO). Because the natural widths (IJ of most of the observed resonances are much less than the target thickness, we do not observe the resonance widths directly; however, some information on the partial widths involved in the reaction can be obtained by measuring the resonance yields. The total cross section for the elastic scattering process in the neighborhood of an isolated resonance will be given by the BreitWigner formula:

where = (2J + 1)/2 is the usual statistical factor, E,, the resonance energy and F g F, + F,, the total width. r,, and r,t are the partial widths in the incoming and outgoing channels, respectively. The yield from a target much thicker than r is then given by gJ

y, = 3

j a(E) dE

(2)

at the peak of the observed resonance, where k is the average stopping power (in keV/g/cm2), A the atomic weight, and N, is Avogadro’s number. The resonance integral is

rr,

i

a(E) dE = G = 2r2X2gJ y

(3)

and we will denote rJ,,/rby r*. Although (2) arises from the uniform energy loss approximation, it should be only slightly affected by straggling, provided that the resonance width is much smaller than the target thickness. The parameter r* approaches whichever partial width is, in a given case, much smaller than the other, since I’, and r,, are the only significant partial widths. Unfortunately, neither partial width is measured uniquely in this way. The appearance of the parameter r* in the peak yield expression (2) is furthermore the mechanism of the channel selection effect of the barrier penetrabilities. The parameter I’* is plainly between 0.5 and 1 times the lesser of the two partial widths, which in most cases we expect to be r,* because of the steep energy dependence of the barrier penetrabilities. The penetration factor PC, (II) is thus a factor in the peak yield, and the rapid decrease of PG* with increasing d’ thus emphasizes channels of small 8’. Assuming for comparison uniform reduced widths yS2 = YD’ = 5 x lo3 eV, we give in Table II the values of I’,Z,, , r (assumed equal to r, + r,e as these are the only important open channels) and the yield parameter

136

KYKER,

BILPUCH,

TABLE ESTIMATES

OF RESONANCE

WIDTI-@

AND

NEWSON

II FOR 2.6-MEV

FQOTONS

ON 50C~

Channel

Err*

(L; J"; 6") (0; l/2-;

1)

(1; l/2-;

1)

(1; 3/2-;

1)

(2)

WI

400

2

400

2

170

10

180

10 20

(2; 3/2+; 0) (2; 5/2+; 0)

35

3U

43

(3; 5/2-; 1) (3; 7/2-; 1)

3

10

13

2

2

(4; 7/2+; 2) (4; 9/2+; 2) a A uniform

reduced

width

m,%

.3

*r

33 50 3 7 1 1

ya of 5000 eV is assumed.

gJ* for the case of 2.6 MeV protons on 50Cr, using values of the penetrabilities Pt taken from the tables of SchilTer (22). These figures show that levels decaying by s- and p-wave protons are strongly favored, others being discriminated against by a factor of 25 or more. On this basis we will assume in our analysis that only those spin states which can give P = 0 or 1 are significant, predominantly the former. The figures in Table II also indicate that the target thicknesses used in the present work (approximately 2 keV) are large compared to the resonance widths. IV.

ANGULAR

DISTRIBUTIONS

AND

SPIN

ASSIGNMENTS

In the present work we have attempted to make spin and parity assignments to the observed compound nuclear levels by means of the angular distribution of the deexcitation gamma rays following inelastic scattering. The necessary formalism has been given by Kraus et al. (13); in the event that a single isolated resonance level of spin J dominates the local yield, the (normalized) gamma-ray angular distribution becomes w(e) = 5 c (-y

d’

c c (2s; + l)l’2 (24 + 1)1’2 c&;,;e c (->y’2 C(2 2 1, -1

s’1 8’a

x W(s;2s;2 : &) W(Js;Js; : 6%) 2(/J/J

Y

: &v) P,(cos 0)

: lJ0) (4)

INELASTIC

PROTON

SCATTERING

137

for a target nucleus whose ground state and first-excited state spins are 0 and 2, respectively. In Eq. (4) 8 and 8’ are the incoming and outgoing proton orbital angular momenta, s’ the outgoing channel spin. The two possible channel spin values 312 and 512 interfere coherently in Eq. (4); 01 is the weighting factor for different channel spin values. The various angular momentum coupling coefficients in Eq. (4) are conveniently tabulated by (for example) Sharp et al. (25). The gamma-ray angular distributions calculated for various spin states from (4) are summarized graphically in Fig. 10. The electric quadrupole nature of the gamma rays observed limits the gamma-ray angular distributions only to terms in cos2(8)

O.!

SUMMARY OF CALCULATED ANGULAF DISTRIBUTIONS

0.:

0.i

0.1

a4

O -0.1

-0.2

-0.3

-0.4

-0.5

-0.6

FIG. 10. Summaryof calculated angular distributions. a2 and q are coefficients of Legendre polynomials; the loci corresponding to various spin states of the compound nucleus are discussed in the text.

138

KYKER,

BILPUCH,

AND

NEWSON

and cos4(B), so that they may be conveniently summarized in this way. The a2 and a4 of Fig. 10 are the normalized angular distribution coefficients, that is as in W(0) = 1 + a,P,(cos 0) + a,P,(cos 0) (where PS and Pa are the Legendre polynomials), predicted in each case. As noted above, the two channel spin modes add coherently in the gamma-ray distribution, so that a family of possible distributions is predicted for a given J”, even assuming only the lowest (and thus most probable) value possible for z?‘. For example, a 5/2- level in the compound nucleus may lead to a distribution with coefficients anywhere on the line labeled A and (3; 5/2-; 1) in Fig. 10; a 3/2- level gives a4 = 0 and 0 < a2 < 0.5, as indicated by the line labeled B and (1; 3/2-; 1). On the other hand, conservation of angular momentum requires that s’, el, and J must in general form a triangle; but for cases where 8’ = 0 we have only s’ = J, and a unique distribution results. For the cases (2; 3/2+; 0) and (2; 5/2+; 0) these distributions are indicated by the open circles so labeled in Fig. 10. The unique distributions are w(e)

=

I +

*P~(co~

e)

oc i +

+P~(COS

e) -

00sye)

(5)

for 3/2+ levels, and w(e)

=

i +

$p,(COS

e) oc i + 6

COS2(e)

- 5

COS4(e)

(6)

for 5/2+ levels. Another unique case is J = 7/2-, L’ = 1, indicated by the cross marked (3; 7/2-; 1) in Fig. 10. Levels with J = l/2 must of course lead to isotropic distributions: these are indicated in Fig. 10 by the square at the origin. However, one factor complicates this simple picture: more than one value of the outgoing orbital angular momentum can in each case occur simultaneously. While the lowest value will be strongly favored by the penetrabilities, it cannot be assumed to be unique. Angular distributions for the different d’ components will be different; thus the distributions can be affected by mixing in /’ as well as in s’. Consider, for example, the case of a 5/2+ compound nucleus level. If 4” = 0 only, the angular distribution is unique, falling at the indicated circle in Fig. 10. If on the other hand the decay were by d-wave outgoing protons only, channel spin mixing would again occur. (Such a decay, i.e., by 8’ = 2 protons only, is of course very improbable). Because mixing both in z?’ and in s’ can occur, a region of Fig. 10 is defined for these mixtures. The unique d’ = 0 point forms one corner of this triangular region; one limit on it is given by line C in Fig. 10, the other happens to coincide with the locus (A) of 5/2- distributions. The shaded region labeled (2; 5j2f; 0) + (2; 5/2+; 2) in Fig. 10 locates the possible distributions for d-wave admixtures less than about 30 %. Similar mixing of 8’ components also occurs for 3/2+ levels. Here all distributions must lie on the a2 axis, and the d-wave component has (for all channel spin mixtures)

INELASTIC

PROTON

SCATTERING

139

negative a, , so that increasing d-wave admixture moves the distributions to the left along the a, axis from the pure (2; 3/2+; 0) circle in Fig. 10. For compound states leading to z?’ > 0 we have shown in Fig. 10, and will consider in analyzing the data, only the lowest possible angular momentum component. This restriction is plausible because the Coulomb barrier penetrabilities discriminate much more strongly between el = 1 and 3 than between d’ = 0 and 2. The degree of uncertainty in assigning spin and parity to the observed levels by means of the gamma-ray angular distribution is suggested by Fig. 10. Thus 5/2levels and 5/2+ levels (with a particular channel spin mixture in the /’ = 2 component) can be confused; a similar uncertainty can exist between 3/2- and 3/2+ levels. In general we see that the degree of uncertainty depends largely on the amount of d-wave admixture which occurs in the 3/2+ and 5/2+ levels. Thus if they decayed exclusively by s-wave protons the distributions from all such levels would be described by the indicated points in Fig. 10 and assignment would be easy. On the other hand, if d-wave admixture can get as high as 50 % considerable confusion may result, especially for levels with J = 3/2. This question can only be answered by the data. All the foregoing discussion has dealt with the angular distribution arising from a pure isolated resonance level in the compound nucleus. However, as is apparent from the excitation functions shown above, there is in every case a significant nonresonant yield of the deexcitation gamma ray on which the fine structure is superimposed. Presumably this arises in part from what we may call “resonance background”-that is, tails of faraway resonances and unobservably weak resonances in unfavored channels-and in part from Coulomb excitation. The former case would involve resonances in many channels. It is pointed out by Kraus et al. that there is no interference between terms of opposite parity in the differential cross section, so that half the resonance background would be expected to add incoherently to an isolated resonance. Terms in the differential cross section arising from interference between two resonances involve the reduced width amplitudes. These may be of either sign and are usually assumed (16) to be of random sign. From this it follows that, in the interference term between one resonance and a background which is a composite of many resonance tails, there will be a great deal of cancellation. On this basis we argue that the part of the nonresonant yield arising from “resonance background” is for the most part incoherent. The contribution, if any, of Coulomb excitation to the observed nonresonant yield is less easy to discuss. The theoretical conclusion (17) seems to be that nuclear and Coulomb excitation contributions to inelastic scattering will combine coherently. On the other hand, a recent attempt to observe effects of such interference in the 23Na( p, p’) reaction (18) gave negative results. Thus it is not possible to say whether a Coulomb excitation contribution to the nonresonant yield observed in this experiment would interfere with a resonance or not.

140

KYKER,

BILPUCH,

AND

NBWSON

The conclusion to be drawn is that we expect most, although perhaps not all, of the nonresonant yield to combine incoherently with the resonances. In order to analyze the angular distribution data on individual resonances we have had to assume that the resonant and nonresonant contributions are entirely incoherent. This is of course incorrect to some degree; however, we cannot improve on this assumption without a more detailed knowledge of the nature of the smooth contribution. Further, this effect is of such magnitude that it should significantly influence only the weakest resonances. The yield and angular distribution of the nonresonant gammas were measured on each target at several energies as far removed from apparent resonance structure as possible. Their intensity and angular distribution were then estimated by interpolation at the energy of each resonance and subtracted as a second “background” from the angular distribution measured on the resonance peak. In a few cases, for example near 2.49 MeV in 50Cr + p, two resonances were found so closely spaced that if the peaks add inchoherently some yield from the lower-energy one would be counted at the maximum of the other. In correcting these cases we must again consider the possibility of coherence. Two resonances may or may not interfere depending on the parity and spin of each, but interference effects will be small if the separation of the two is several times the natural width of each resonance. The closest spacing we see in such cases is about 2 keV. From the estimates of Table II it thus seems unlikely that the widths of 3/2+ or 5/2+ resonances will be large enough to give significant interference. On the other hand, if a given resonance appears wider than the target thickness (as does, for example, the 2.702 MeV resonance in 4”Ti + p) it is probably a negative-parity state, i.e., formed by P-wave protons, in which case it would not interfere with a positiveparity resonance, On this reasoning we have felt it sound to neglect the possibility of interference between adjacent levels and have subtracted such cases accordingly. For each target the resonance shape appears, with a few exceptions, to be quite uniform (a further indication that in most cases the natural widths are much smaller than the target thickness); this makes it possible to estimate fairly accurately the amount of one resonance which must be subtracted at the peak of another. When these corrections had been made to the angular distribution data on each resonance, the resulting data were fitted with a curve of the form N[l + a,P,(cos 0) + a,P,(cos 13)]. The curve-fitting was done using a general linear regression program made available by the Duke University Digital Computing Laboratory. The normalized angular distribution coefficients a2 and a4 are plotted on a graph such as Fig. 10; the spin and parity assignment is estimated from the position of each resonance point relative to the loci of Fig. 10.

INELASTIC

PROTON

V. RESONANCE

141

SCATTERING

PARAMETERS

Angular distribution data has been taken on essentially all resonancesobserved in 46Ti(p, p’) and 50Cr(P, p’). A very few weak resonanceshave been omitted. In the 56Fedata about one peak in six wastoo weak or too closely entangled with other structure to be analyzed in this way; and in the case of 48Ti only the strongest resonances seen in Fig. 5 were investigated. The properties of all resonances analyzed are summarized in Table III. Column 1 of Table III gives the number of each resonance as indicated on the excitation function plots; columns 2 and 3 the laboratory proton energy and the excitation energy of the compound nucleus level, respectively. Excitation energies were calculated from the massvalues given by Everling et al. (19). Columns 4 and 5 give the normalized angular distribution coefficients a2 and LQ, corrected for finite angular resolution; column 6 the spin assignmentinformation deduced from them. Assignments given first in column 6 are to be preferred to alternatives given in parentheses.As the parity ambiguity for f = 312and 512statesis largely due to the possibility of D-wave admixture in the outgoing protons for the positive-parity case,we show in column 7 the percentageof c’ = 2 decay necessary,on the assumption of positive parity, to explain the observed distribution. Column 8 gives the resonance integral (3 defined in Eq. (3). In calculating this parameter for each resonancethe calculations of Vegors et al. (20) on absolute efficiency of scintillation detectors were used. This calculation is probably the least precise aspect of the present work, primarily because of the large systematic uncertainty in current integration calculation which is estimated to be about h25 ‘A. This situation is worse for the Ti targets where we did not measure angular distributions against

0

FIG. 11. Gamma-ray 5octiP, P’).

ANGULAR

DISTRIBUTION

078

Me” ?’ YIELD

I 309

angular

OF

I 66

distribution

I 90’ e PY

measured

I 120’

I 150’

on the 2.271-MeV

I

resonance

in

4’V

Compound Nucleus

1 2 3 4 5 6 I 8 9 10 11 12 13 14 I5 16 17 18 19 20 21 22 23

Res. Number

2.506 2.510 2.521 2.529 2.549 2.558 2.567 2.578 2.592 2.597 2.612 2.623 2.635 2.638 2.651 2.678 2.682 2.687 2.692 2.696 2.703 2.707 2.717

7.644 7.648 1.659 7.667 7.686 7.699 7.708 7.717 7.738 7.743 7.758 7.769 7.780 7.783 7.796 7.812 7.816 7.821 7.826 7.832 7.835 7.839 7.849

SUMMARY

III

.13* .03 .34* .03 .41* .03 .42+ .03 .23+ .07 .54* .05 .47+ .03 .61’ .04 .53* .03 .10’.04 .57+ .04 .59& .04 .30+ .03 .70+ .05 .57* .04 .37* .03 .50’ .02 .23’ .lO .43& .04 .27* .05 .20* .02 .5@ .02 .61* .02

0 .52* -.48* -.13* 0 0 0 -.56-L0 0 -.61+ -.55’ 0 -.61* -.53+ 0 0 -.30* - .43* .09* 0 - .47* -.55* .02 .02

.12 .04 .06

.05 .04

.05 .04

.04

.03 .03 .03

PROPERTIES

Angular Distribution Coefficients (4 (4

OF RESONANCE

TABLE

312 (5/2+) 5/2+ 5/2* 312 312 312 5/2+ (5/2-) 312 312 5/2+ (5/2-) 5/2+ (5/2-) 312 5/2+ (5/2-) 5/2+ (5/2-) 312 312 5/2+ 5/2+ 5/2* (312) 312 5/2+ (5/2-) 5/2+ (5/2-) 5/2-

Spin Assignment % 55 90 25 40 40 0 5 0 0 60 5 5 30 0 0 20 0 55 22 68 40 5 0

D (+) 6.5 8.3 3.9 10.7 6.1 4.5 28. 31. 18. 3.8 10.3 14.0 3.7 2.8 9.3 4.6 12.8 16. 7.8 3.3 47. 52. 5.4

(ba:-eV)

6.5 5.7 1.2 2.2 3.0 1.2 .6 2.0 1.5 4.2 3.5 1.7 1.1 15.5 11.5 1.2

1.9 1.4 8.7

.8 2.2

r* (eV)

;fi

5

E F s “=

2.734 2.750 2.759 2.783 2.788 2.796 2.812 2.818 2.826 2.830 2.832 2.838 2.840 2.844 2.853 2.862 2.868 2.871 2.879 2.898 2.906

2.681 2.687 2.690 2.697 2.705 2.709 2.714 2.737 2.745

24 25 26 27 28 29 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9

9.380 9.386 9.389 9.396 9.404 9.408 9.413 9.435 9.443

7.865 7.880 7.890 7.914 7.919 7.927 7.942 7.948 7.955 7.958 7.960 7.965 7.967 7.971 7.980 7.989 7.994 7.997 8.005 8.023 8.031

0

.03 .04 .04 02 .03

.02 .04 .03 .03 .02 .04 .02 .04 .04 .04 .03 .04 .03 .04 .Ol .04 .Ol .04 .03 .Ol

0 .12* .07 .47+ .03 .42* .02

.52* .48* .36* .45* .47*

.45+ .23f .33* .08* .63* .41* .36’ .47* .19* .67* .58* .30* .28f .17-t .07+ .67” .11* .37’ SW .14* - .57+ 0 .07* 0 -.17* 0 0 - .47* 0

0 - .55* 0 0 0 0 -ho* - .48& 0 .15* - .34+ 0 - .56* 0 - .34+ 0 0

--.18*

0 0 0

312

.03

312 5/2+ .03

512’ (5/2-) 312

112 312

5/2*

312

5/2+- (5/2-)

312, 112 312

5/2+ (5/2-) 312 5/2+

312

312 S/2+ (5/2-) 5/2+ (5/2-) 312 5/2+ 5/2+

312

312 312

5/2+ 312 S/2+ (5/2-)

.04

.03

.04

.04

.03 .05

.04 .03

.02

.03

l/2 312

55 15 10

7 0 55 5 5

5 40 45 60 0 12 20 5 45 0 5 30 70 60 60 0 55 35 65 50 22. 4.6 6.8 12.7 26. 7.6 9.0 11.2 14.6

25. 4.8 20. 3.1 3.5 41. 2.9 21. 15.1 34. 16.5 14.3 8.9 25.9 8.2 130. 24. 86. 14.5 45. 135.

1.5 1.5 4.2 5.8 5.0 3.0 2.5 4.9

4.8

15.8 47.9

45.6 5.7 30.0 3.4

8.5 1.6 6.7 .7 1.2 9.3 1.0 7.2 5.2 11.8 3.8 3.3 3.1 6.0 1.9

3

x s 3 Et

g El 2

z F % 2 Q

61Mn

Compound Nucleus

2.162 2.183 2.214 2.217 2.224 2.270 2.298 2.319 2.329 2.340 2.349 2.399 2.404 2.455

2.767 2.773 2.783 2.795 2.803 2.806 2.810 2.821 2.823 2.827

10 11 12 13 14 15 16 17 18 19

1 2 3 4 5 6 7 8 9 10 11 12 13 14

(MeV)

E

Res. Number

Table ZZZ (contimed)

7.406 7.425 7.457 7.460 7.466 7.510 7.537 7.557 7.567 7.578 7.587 7.639 7.644 7.693

9.465 9.471 9.481 9.492 9.500 9.503 9.507 9.518 9.520 9.524

.29* .55* .38’ .64* .44* .48+

.29+ .24” .24* .53* .51* .53* .25+

.55+ .59+ .26* .40* .51+ .51* .42+ .45& .43* .42*

0 .02 .02 .05 .04 .06 .02

.06 .06 .04 .05 .04 .05 .04

.03 .05 .06 .03 .02 .04 .02 .05 .03 .05 0

.03 .05 .06

.06 .06

- .62* .05 -.60* .04 0

- .50+ .02 0

0 0 - .47* .04 -.51* .05 0 0 0

-.13* -.31*

- .34& .05

- .49* .02 -.49-t .02 0 0 .09+ .03

-.56* -.55* -.43*

Angular Distribution Coefficients (4 6%)

312 5/2+ (5/2-) 312 5/2+ (5/2-) 5/2+ (5/2-) 312

l/2

312 312 5/2+ (5/2-) 5/2+ (5/2-) 312

5/2+ 5/2+

5/2+ (5/2-) 5/2+ (5/2-) 5/2+ 312 5/2+ (5/2-) 5/2+ (5/2-) 312 312 5/2+ 5/2+

Spin Assignment ‘A

30 5 15 0 10 0

55 50 35 0 10 10 35

0 0 40 15 10 10 10 5 55 30

D (+>

3.4 3.9 3.0 4.1 5.0 3.2 8.5 9.7 70. 97. 6.3 5.6 4.6 29.

31. 8.9 9.7 16.7 24. 20. 12.8 8.4 11.3 5.6

6 (bam-eV)

.9 .6 2.4 5.5 19.8 18.5 1.8 1.1 .9 8.6

.6 .7 .8 1.1

1.3

4.4 2.9 2.6

4.5

6.9 2.0 2.2 5.7 5.5

3

$

5

s “E

E G

F

15 16 17 18 19 20 21 22 23 24 25 26 28 29 30 31 33 34 3.5 36 37 38 39 40 41 42 43 44 45 46 47 48 49

2.470 2.492 2.495 2.506 2.518 2.537 2.540 2.549 2.554 2.566 2.570 2.592 2.602 2.624 2.629 2.634 2.668 2.681 2.689 2.696 2.709 2.728 2.739 2.750 2.754 2.763 2.768 2.774 2.782 2.792 2.806 2.823 2.833

7.708 7.729 7.732 7.743 7.755 7.774 7.777 7.786 7.790 7.802 7.806 7.827 7.837 7.859 7.864 7.869 7.913 7.926 7.933 7.939 7.953 7.972 7.983 7.993 7.997 8.006 8.015 8.018 8.024 8.034 8.048 8.065 8.074

.56' .04 .49+ .02 .49+ .02 ..56* .03 .12+ .03 .09* .04 .25* .06 .39-t .03 .46* .02 .43* .05 .35* .04 .4s* .08 .38+ .06 .63-t .04 .53* .02 .38-t .04 .49+ .08 .15+ .03 .39+ .02 .05+ .02 0 0 .51* .06 .12* .03 0 .51+ .02 ,581 .04 .33-t .05 .oa* .Ol .26* .Ol .53' .02 .17+ .03 .54* .02

--.61* .05 0 0 0 -.05* .03 -.07-t .04 -&I* .07 0 0 ~ .37+ .06 .53* .04 -.49* .lO .12* .07 - .56* .05 0 .11* .04 -.23+ .09 0 0 0 --.16* .04 0 - .60- .07 0 0 0 - .53* .05 .151 .05 0 0 - .62' .03 -.31* .03 - .62* .02 10 55 0 312 512+ (5/2-) 5/2- (5/2+) 312 312 5/2+ (5/2-) 512 i5/2+ (512-j

112

112 5/2+ (5/2-) 312

60 35 0 60 0

&

(6’3 0 0 (60) 30 50 20 60 85

45 20 10 30 (85) 20

5/2+ 312 312 5/2', 7/25/2- (S/2+) 5/2+ (5/2-) 5/2- (5/2+) 5/2+ (5/2-) 312 5/2-- (5/2+) 5/2+ (7/2-j 312 312 312 5/2+

W+ U/2)

512 ' (5/2-j 312 312 312 5/2+ (3/2)

14.9 39. 35. 44. 20. 6.8 4.8 24. 8.3 2.9 23. 3.8 2.8 12.2 26. 12.1 6.4 8.2 89. 45. 4.5 5.2 5.8 68. 19.2 46. 26. 11.5 45. 37. 31. 11.3 23.

3.0 11.9 10.7 13.5 6.1 2.1 1.0 7.4 2.6 .6 4.9 .8 .6 2.6 8.3 2.6 1.4 2.7 29.3 14.8 1.0 3.5 1.3 22.7 12.9 15.4 5.8 2.6 15.4 12.6 7.0 2.6 5.2

1

%o

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Res. Number

Compound Nucleus

Table ZZZ(continued)

2.607 2.617 2.625 2.629 2.632 2.648 2.657 2.660 2.676 2.683 2.690 2.700 2.718 2.724 2.732 2.737 2.745 2.755 2.763 2.768 2.780

8.851 8.860 8.868 8.872 8.875 8.890 8.900 8.902 8.918 8.925 8.932 8.941 8.959 8.965 8.973 8.978 8.986 8.995 9.003 9.008 9.020

(MeV)

E*

40-t .29* .34* .58* .49* .67* .36’

.55* .49* .47* .53* .51* .54’ .52* .63* .30* .46* .51* .14* .46* 0 .03 .05 .03 .04 .05 .08 .04

.05 .04 .05 .04 .04 .05 .06 .06 .05 .05 .04 .02 .04

-.51’ -.51* - .26* 0 - .58* - .34* ~ .50* -.51* - .45* 0 - .49* - .22’ - .48’ 0 0 0 0 0 - .65* - .47+ 0

Angular Distribution Coefficients (4 (4

.05 .09

.05 .03 .05

.04 .06 .06 .07 .05

.06 .05 .05

312

312 312 312 312 5/2+ (5/2-) 5/2+ (5/2-)

112

5/2+ (5/2-) 5/2+ (5/2-) 5/2+ 312 5/2+ (5/2-) 5/2*, 7/25/2+ (5/2-) 5/2+ (5/2-) 5/2+ 312 5/2+ (5/2-) 5/2+ 5/2+ (5/2-)

Spin Assignment y0

15 30 20 0 10 0 20

5 15 30 0 5 30 10 0 35 5 10 70 15

D (+)

2.9 6.6 2.8 4.4 6.1 2.8 4.6 2.8 3.2 9.2 5.0 9.6 5.4 3.8 18.6 3.3 3.6 15.4 6.7 1.8 8.5

IT (bam-ev)

&

3.0 1.1 2.1 1.2 2.5 6.2 1.1 1.2 5.2 1.5 .4 2.9

.8

1.4 1.3 .6 1.0 .6 .7

2

5 Q

m F z

.6 1.4 .6

? E

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 47

2.786 2.198 2.801 2.812 2.822 2.829 2.844 2.850 2.852 2.851 2.860 2.866 2.814 2.817 2.886 2.888 2.898 2.906 2.912 2.917 2.922 2.930 2.933 2.940 2.948 2.952

9.026 9.037 9.046 9.051 9.061 9.068 9.083 9.089 9.091 9.095 9.099 9.106 9.112 9.115 9.124 9.126 9.136 9.145 9.150 9.155 9.160 9.167 9.171 9.177 9.185 9.189 .02 .02 .04 .05

.42* .43* .24* .25’ 0 .54* .02 .28* .06 0 .46* .04 .56* .05

.05 .03 .04 .05 .09 34 .04 .02 .04 .04 .05 44 .03 .02 .02

.53-t .47* .41* .34+ .53* .47+ .56* .5w .58* .56& .36* .54* .48+ .46’ .48+ .04 .04 .03

.04

.04

.lO

0 .06* .02 0 0 0 -.56*.03 0 0 0 --.61*.06

0 - .58+ 0 -.56* 0 0 - .58* - .54* --.50* 0

-.38*

0 0 0 0

5 0 312 5/2+ (5/2-)

112

5 30

35 35

10

0 5 10 20 30 5 0 0 0 0 20 5 10 15 0

5/2+(5/2-) 312

l/2

312 312, 5/2+ 312 312

312 312 312 312 5/2*, 7/2312 5/2f (5/2-J 312 512’ (5/2-) 312 312 5/Z+ (5/2-) 5/2+ (5/2-) 5/2+ (5/2-) 312

5.3 9.4 9.3 3.8 3.0 21. 5.6 35. 12.9 11.2 6.3 18.9 18.4 32. 26. 11.6 21. 58. 11.0 2.5 6.2 19.3 5.0 11.0 10.6 9.6

1.8 3.2 4.0 1.3 .7 1.4 1.3 12.3 3.0 3.9 2.2 4.4 4.3 1.4 9.1 4.1 7.4 20.5 3.9 .9 4.4 4.6 1.8 7.9 3.8 2.3

148

KYKER,

BILPUCH,

I

I

AND

NEWSON

I

I

I

I go0 8 PI

I 120’

I 150°

t [[

’ 1

[r

0’

FIG.

ANGULAR

DISTRIBUTION

0.76

MeV Y YIELD

I 30*

12. Angular distribution

I 60’

OF

measured on the 2.495MeV

resonance in %r(p,

p’).

integrated beam. The degree to which the incident beam spread depressesthe observed peak height can be estimated from the resonance shapecalculation mentioned above, and the stated values of 0 have been corrected for this effect. Column 9 gives the partial width combination P = I’,I’,r/r, which comes directly from the resonanceintegral 6. Two typical angular distribution measurementson 50Cr are shown in Figs. 11 .and 12. Here the plotted curves are not the best fits found by the regressionprogram but the theoretical shapes[Eqs. (5) and (6)] for the pure (2; 5/2+; 0) and (2; 3/2+; 0) processes, respectively; however, the differences are negligible. Figure 1 is interesting in that it shows data on a very weak resonance. The degree to which the experimental point scatter on the weakest resonancesis reasonable is an indication of the reliability of the background subtractions which have been made. With few exceptions the data on weak resonancescompare favorably (except for generally poorer counting statistics) with the data on the stronger ones. The angular distribution data on 46Ti, 50Cr, and 56Feare summarized in Figs. 13-15. In these plots a point for each resonance is superimposed on the scheme of predicted angular distribution coefficients shown in Fig. 10. Error bars shown in these figures are those produced by the regressionprogram; that is, they indicate the quality of the fit without any reference to absolute counting statistics, etc. One rather disappointing aspect of the 46Ti data is evident in Fig. 13; there is apparently some small systematic error in this data which has shifted the group of points near (0.57, -0.57) somewhat to the right. Resonancesboth large and small are included in this “shifted” group so that it is not possible to understand this effect as a consistent background subtraction error. As yet no satisfactory explanation for

INELASTIC

PROTON

149

SCATTERING

this discrepancy can be given. Apparently it is something specific to the 46Ti data as no such systematic discrepancy is evident in Figs. 14 and 15. We see in Figs. 13-l 5 illustrations of the ambiguity of spin assignment discussed above. As pointed out previously, in most cases a definite distinction between J” = 3/2+ and 3/2- or between 5/2+ and 5/2- cannot be made for each resonance. However, in all three cases the clustering of points near (0.57, -0.57) and, to a lesser extent, near (0.5,O) are visible as is to be expected if 3/2+ and 5/2+ interaction is dominant. An occasional point must be assumed in error by twice the limits plotted to be assignable or even possible; however, these error limits, generated by the regression program, can be unreasonably small in case of a fortuitously good fit. Q4 + 0.4

P,y DISTRIBUTIONS Ti4’ (p,p,yO***)

2.50 to 2.90 MeV

:

\

a2

0.6--

t

t

FIG. 13. Summary of the gamma-ray angular distributions measured on resonances in 4BTi(p p’). Each point corresponds to the angular distribution at a single resonance; the error bars ake the uncertainties in the coefficients a2 and a4 found by the fitting program. i

150

KYKER,

BILPUCH,

AND

NEWSON

a4 +

0.4

p,y

DISTRIBUTIONS

CrSo(p,p, 2.15

t

to

y

a7a)

2.85

MeV

:

t FIG. 14. Summary of experimental angular distributions VI. SPIN OF THE FIRST-EXCITED

at resonances in Wr(p,

STATE

OF

p’).

“CR

Our use of (p, pl) angular distributions as a means of assigning spin and parity to compound nucleus levels has depended throughout on the assumption that the residual excited state is of spin and parity 2+. For the target nuclei 46Ti, 48Ti, and 56Fe this assignment is well established (29, but little information exists about the first excited states of 50Cr and 54Fe except that the absence of beta transitions from the Of ground states of 60Mn and 54Co argue against a 0+ assignment for either of these states. (Beta transitions involving no change of either spin or parity are allowed on the Fermi selection rules.) Certainly both the predictions of simple models for the low-lying states of even-even nuclei (22), and their systematics, favor a 2f assignment very strongly; however, specific evidence for assigning these states 2+ is lacking.

INELASTIC PROTON SCATTERING

151

In the case of 5oCr, we have measured many y-ray angular distributions from the 0.78-MeV first-excited state, and these observations should reveal the spin (j,) of this state as well as that (in each case) of the compound nucleus 51Mn. First, the gamma transition 0 ---f0 is completely forbidden (a photon must carry off at least one unit of angular momentum), so that the mere existence of the 0.78-MeV transition rules out j, = 0. Furthermore, the multipolarity (L) of the transition will limit the complexity of the angular distribution (23) to COSTS. Since the final state has zero spin, we have a pure multipole transition with L = j, . Thus the strong cos4(0) dependence observed in many of the measured distributions (see Table III) rules out j, = 1. The complete absence of apparent COY?dependence in the data suggests j, = 2, but is not conclusive as J and G likewise limit the complexity of the angular distria4 p,y

DISTRIBUTIONS

Fe56 ( P,P, Y o.64) 2.60 to 2.95MeV

a2

FIG. 15. Summary of experimental angular distributions

at resonances in 66Fe(p, p’).

152

KYKER,

BILPUCH,

AND

NEWSON

bution [see Eq. (4)] and low values of / are emphasized by the barrier penetrabilities. However, the value of j, certainly places restrictions on the observed angular distributions. Barnard et al. (24) give a formula for the angular distribution coefficients in terms of the fractional populations p(mJ of the magnetic substates of the gamma-emitting state. Using their result one finds that the many observed distributions with strong cos4 terms but little or no cos6 dependence are not consistent with any possible set of p(mJ if j, = 3 for 50Cr. Against the possibility of still higher spin for the first excited state of 5oCr a more direct argument can be invoked. If this transition were 4 + 0 the single-particle estimate of its lifetime (25) would very long; whereas if j, = 2 the estimate is around lo-i0 second or less, and if the transition is E2 the transition probability is almost certainly enhanced by collective motion. For 46Ti, 4sTi and 56Fe, the corresponding experimental values (26) are -IO-11 sec. However, if the lifetime of the first excited state of 5oCr were anything like that implied by the assumption of j, > 4, one would expect that the orientation of the nucleus in this state would be completely destroyed by interaction with extranuclear electromagnetic fields, and that the strongly anisotropic distributions observed in 5oCr(p, ply) would not occur. In the Coulomb excitation of tungsten by -3.5MeV protons (27), as much as a 50% attenuation of the anisotropy could be attributed to this cause; in that case the lifetime of the gamma-emitting state was estimated as ~10~~ sec. From these considerations we conclude that the 0.7%MeV first-excited state of 50Cr has spin 2; and, given this, the assumption of positive parity for this state from the systematics of even-even nuclei is much strengthened. Thus, the first excited states of all the targets (46Ti, 48Ti, 5oCr, and 56Fe) on which we have measured gamma-ray angular distributions can be taken with assurance as having spin and parity 2+. VII.

LEVEL

SPACINGS

Level spacings in highly excited nuclei have been extensively investigated in this laboratory. This emphasis can be attributed to the relative ease of treatment of this quantity, both experimentally and theoretically. Recently considerable success has met the efforts of Newson and collaborators (1)-(J) to understand quantitatively the detailed variation of l/2+ level spacings in compound nuclei in the mass region we are studying. These authors have shown that, independent of the effects of excitation energy, the level density is strongly dependent on the shell structure of the target nucleus in the manner predicted by Rosenzweig (28) from considering the effects of the gaps at the magic numbers in the shell model level scheme. There are also indications of a dependence on neutron excess, as predicted theoretically (29) (30). It is interesting to see whether our data support these conclusions.

153

INELASTIC PROTON SCATTERING

In our discussion of relative level spacingswe will concentrate on 5/2+ levels as these are more easily indentified experimentally than are 3/2-t levels. We must first consider how accurately the present data determine the 5/2+ level spacings. Two considerations enter here, which to some degree compensate one another: levels of spin and parity 5/2- or 7/2- may be incorrectly assignedas 512%by angular distributions; on the other hand it is certain that some of the weakest 5/2+ levels have been missed. Now in estimating the degree of 5/2- contamination we are aided by the fact that the angular distribution from a 7/2- level, at least for the dominant r/’ = 1 term, should be unique. Referring to Figs. 13-15, which summarize the angular distributions measured on the targets 46Ti 50Cr and 56Fe, we find a total of four resonanceswith approximately the angular distribution predicted for a 7/2- compound nuclear state. Arguing that 5/2- and 7/2- levels should be observed in roughly the samenumber and strength, we conclude that a fair estimate of the average number of 5/2- and 7/2- levels observed in each compound nucleus would be three, or about 15%, in each case, or the resonanceswhich could be assigned5/2-‘-. The manner in which the local level spacings(D) are distributed about the mean values (D) in each compound nucleus gives another indication of the number of 512~.7/2-, or other levels being included in the 5/2+ count. The preponderance of experimental evidence supports the Wigner (31) distribution, which predicts a scarcity of very close spacings(“level repulsion”). If levels of other spin states are being included in the 5/2+ sample, however, these will be distributed in a random manner, as the level repulsion occurs only within a single spin state. Lane (32) gives the formula for combining two spacing distributions of arbitrary form. In Fig. 16 we show the experimental spacing distribution in comparison with (a) the I

Distribution

/

of 54tLevel

Spacings

-

FIG. 16. Distribution of 5/2+ level spacings. The points represent the experimental distribution; 63 spacings are included. The dashed curve is the Wigner distribution; the full curve is the Wigner distribution corrected for an assumed 20 % of the resonances wrongly assigned.

154

KYKER,

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Wigner distribution and (b) the corrected distribution corresponding to 20 % of the included levels being other than 5/2+. Plainly the latter curve is the better fit to the data. Although the experimental distribution is not good enough, statistically, to enable us to use this calculation alone as a quantitative determination of the proportion of wrongly assigned levels, Fig. 16 does support the approximate estimate of 15 % arrived at previously. Now let us consider the likelihood of our having missed some of the weakest resonances. As will be discussed below, the width parameter r* quoted for each resonance in Table III cannot be identified with a unique partial width; thus we cannot make use of the reduced width distribution to estimate the number of weak resonances missed. We have, instead, estimated level spacings in the compound nuclei *‘V, 51Mn, and 57Co from the slopes of integral plots of the estimated number of 5/2+ levels against energy. If we are missing weak resonances, it is reasonable to expect that more will be missed at lower energies in each case, where the penetrabilities are lower by a factor of from 5 to 20 than at the highest energies; this would be reflected as a lower slope at lower energies, that is a curvature, in the spacing plots (not shown). This is found to be the case for 51Mn, as we ran with the 50Cr target to energies considerably lower, compared to the Coulomb barrier height, than with the other targets. For the compound nuclei *‘V and 57Co, on the other hand, the plots are very good straight lines. The observed level densities are taken to be the slopes of these lines (above 2.5 MeV for the 50Cr data), corrected for the proportion of wrongly assigned levels estimated as discussed above. Finally, as it was experimentally necessary to omit angular distributions on about 15 % of the resonances observed with the 5sFe target, we have corrected accordingly the level spacing in the compound nucleus 57Co derived from the plot. While we recognize that our handling of the level spacing data is necessarily approximate, it is unlikely in the extreme that we have not been accurate enough to draw some reliable statistical conclusions. The resulting mean spacings (D) of 5/2+ levels are those shown in Table IV. All things considered, the precision of these numbers can probably be taken as i-25 %. TABLE S/2+ LEVEL

Compound Nucleus 47V “‘Mn 6’co

N

Z

24 26 30

23 2.5 27

IV

SPACING

DATA

D WV)

mean (MeV

20 30 17

7.85 7.75 9.05

E*

1) (8 MeV) (kev) 17 24 40

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In order to consider the systematic behavior of the level spacings we must correct to a uniform excitation energy (arbitrarily chosen as 8 MeV). Over a small energy range we may use the “constant temperature” approximation for the energy variation of the level density, f@*)

= ~(0) ew(E*lT);

T = (dln p/dE*)

= constant,

(7)

where p = (D)-l is the level density and T the nuclear temperature. Little data on nuclear temperatures in this region are available. Bowman (2) found an average value of 1.2 MeV appropriate for neutron level densities in this mass region, and the same average temperature follows from the Fermi gas model (33) level density coefficients measured by Kapadia (34) in neutron inelastic scattering. We have accordingly put T = 1.2 MeV in Eq. (7) and corrected all the level spacings in Table 4 to 8 MeV excitation energy. The Duke experiments (Z)-(3) on neutron level spacings in this mass neighborhood have suggested two systematic effects. Strong fluctuations in the Ievel spacings (corrected to common excitation energy) are observed which are closely correlated to the shell structure of the target nucleus. The models of Rosenzweig (28) and of Newson and Duncan (35) both reproduce these shell effects quite satisfactorily. There are also indications of a systematic decrease in level spacing with increasing neutron excess. A summary of this analysis as applied to 27 targets is given by Farrell ef al. (3), who find that, at these energies, the major shell closure appears at 16 instead of 20 particles. In the present work we have reliable measurements of 5/2+ level spacings for only three nuclei. Clearly this is insufficient for the kind of detailed analysis carried out for the neutron experiments, or for isolating the various effects dicussed above. However, there is a variation in the level spacings (at common excitation energy) among the three compound nuclei in Table IV which is consistent with the shell structure dependence found in neutron level spacings. 57Co, with 27 protons and 30 neutrons, is closer to being a doubly-magic (N = 2 = 28) nucleus than either 47V or 51Mn. Correspondingly, the 5/2+ level spacing in 57Co is twice as large as in the other two. Thus all the variation of B(8 MeV) observed in these three nuclei can be accounted for by the kind of shell structure dependence discussed by Farrell et a I. In Table III there appear roughly the same number of levels assigned 3/2 as 5/2+. Certainly one expects fewer 3/2+ than 5/2+ levels on the basis of the formula p(J) =

p@W+

I)kw[--J(J

+

1>/2u21)

(f-3)

derived from statistical theory (30). However, some 3/2- are certainly included in the 3/2+ category, as are perhaps a few levels of J > 312 with a fortuitous channel spin mixture. An average of two or three resonances, presumably l/2-, with

156

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isotropic angular distributions are seen in each nucleus studied. From Eq. (9) we would expect about twice this many 3/2- levels to be observed, and about 50 % more 5/2+ levels than 3/2+ levels. Within experimental uncertainties, therefore, our data are consistent with Eq. (8).

VIII.

STRENGTH

FUNCTIONS

Another average property of resonances of considerable interest is their strength or average width in a given reaction channel. This is expressed in terms of a “strength function” S = (y2/D>, the average ratio of reduced partial width to spacing for a given channel. The reduced widths (as defined by Lane and Thomas (II) are used to eliminate the rapid energy dependence of the penetration factors. In the last several years the 4 and P-wave neutron strength functions have been studied intensively, primarily by means of neutron total cross section measurements (31). The main features observed are fairly well accounted for by the predictions of the complex potential well (optical) model of the nucleus, the most prominent being the maxima observed in the low-energy neutron strength function near A = 55 and 160 for S-wave neutrons and near A = 100 for P-wave neutrons. The situation with respect to charged particle strength functions is considerably less well in hand. Margolis and Weisskopf (38) pointed out that the nucleus is a somewhat shallower well to protons than to neutrons due to the Coulomb interaction, so that strength function maxima should fall at somewhat higher mass numbers for protons. For example, they predict the 3s peak for low-energy protons near A = 70 rather than 55. This is borne out to some degree by the results of Schiffer and Lee (39), (40) on the average yields of (p, n) and (p, p’y) reactions in this mass region, which indicate a peaking at A - 65 and another, less pronounced, at A - 50 which is interpreted as a D-wave strength function maximum. As was seen above, we are able to deduce from the observed yield at the peak of a resonance a parameter r* = rJ,p/r, which is a combination of the partial widths I’, and r,, involved in the reaction cross section. If the penetration factors Pe are such that, in a given case, one can assume r, > r,, [as, for example, in the case of 5sNi(p, p’y) investigated by Schiffer et al. (36)], then r* E r,, . Unfortunately, as indicated by the estimates in Table II, the two penetrabilities which enter the most probable cases (/ = 2, d’ = 0) are, in the present data, of the same order of magnitude, so that it is not possible to identify r* with one or the other partial width. From the data summarized in Table III we cannot, then, derive directly the strength function in either channel; but the tabulated parameter r* certainly contains some information on the subject. If, as is the case for the three nuclei

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most completely analyzed here, the two penetration factors P,,(E’) and P,(E) which enter for the most probable channels are nearly equal, we can define

where S(p) and S(p’) are the strength functions according to the usual definition [S(p’) = (yi,/D>, etc.] in the elastic and inelastic channels, respectively. Thus for 3/2+ or 5/2+ levels S(p) is the D-wave proton strength function, S(p’) the S-wave strength function. The quantity S* plainly bears the same relation to the two strength functions as does r* to the two partial widths, and thus is half of either if the two are approximately equal, and so forth. The quantity S* is the extent of the strength function information which can be extracted from the present data. Values of S* extracted for 3/2f and 5/2+ levels in the compound nuclei 47V, 51Mn, and 57Coare shown in Table V. For easeof comparison the ratio of each S* to the strength function value associatedwith the “black nucleus” model (41) is also given. TABLE REDUCED

Compound Nucleus

4’V 51Mn S’CO

STRENGTH

V FUNCTION

RESULTS

Ratio of S* to Black Nucleus Value

Spin State

S*

3/2+

(1.8 h .6) x 1O-2

.41 f

5/2+

(1.5 f

S) x 10-Z

.34 f

.11

3/2+

(7 It 1.5) x IO-2

1.7 *

.7

5/2+

(4 * 1.5) x IO-2 (3.6 + 1.3) x 1O-2

.80 *

3/2+ 5/2+

(1.8 i

.6) x 1O-2

.15

1.0 + .3 .31

.42 31 .13

The rather large absolute errors indicated in Table V are estimated from integration accuracy, resonance statistics, and negative-parity competition. The latter is the source of considerable uncertainty in the figures for 3/2+ levels. As pointed out previously, large D-wave admixtures are perfectly possible for these even-parity resonances, but more likely to occur on small resonances than on strong ones. Thus the very strong resonanceswith a, = 0 and small a2(which occur in the 46Ti and 50Cr data) are more likely to be 3/2- than 3/2f and have been weighted accordingly in figuring S*. Again, the isotropic resonancesseen are most likely l/2- levels, and the strength with which these occur can afford an estimate of the 3/2- strength function. Finally, somefew resonanceswhich could be assigned3/2f

158

KYKER,

BILPUCH,

Ep (lob)

AND

NEWSON

(t&V)

FIG. 17. Vo strength function plots. The histograms show the sum of the reduced resonance widths I’*/2P,,(E’) for (A) 3/2+ and (B) S/2+ levels. The slopes of the lines drawn through the histograms are taken as the experimental values of the strength function combination S*.

are noticeably wider than the rest and thus have r s 1 keV or more; the figures of Table II indicate that such levels are 3/2-. The figures of Table V were computed by plotting r*/2P,(E’) against energy for each channel, weighting or eliminating a few of the J = 312 resonances according to these considerations, and drawing the best straight line through the resulting histogram. The slope of this line was taken as S* in each case. The plots for 3/2+ and 5/2+ states in 57Co are shown in Fig. 17. It is immediately obvious from Table V that S* is larger, in 51Mn and in 57Co, for 3/2+ levels than for 5/2+ levels, even after we have tried to correct for 3/2states. It should be noted that, due to the lack of resonances in any strength falling near the 7/2- point in Figs. 13, 14, and 15, we have neglected the possibility of 5/2competition in the calculations for 5/2+ resonances, except that a few resonances which would require very large D-wave admixtures to explain on a 5/2+ assignment have been weighted somewhat less than 1. Any appreciable 5/2- competition would simply enhance the difference between channels which is apparent in Table V.

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159

This difference between spin states in the value of S* we believe therefore to be definitely indicated in spite of the difficulties involved in calculating it. And this is not difficult to understand. As noted above, there are both experimental and theoretical indications of a maximum in the D-wave proton strength function near A = 50; there is, furthermore, additional evidence for the existence of this peak in the present data, as will be discussed below. Now in recent years there has been considerable study of the A LY 100 peak in the P-wave neutron strength function, and presently the balance of the evidence points to a splitting (or at least a widening) of this peak, most easily explainable by the inclusion of a spin-orbit term in the optical potential (37), (42), (43). There is other evidence for the existence of such a term, although its ability to cause the observed splitting is somewhat doubtful (44) (45). There is every reason to believe that the same effect should be present in the optical potential for protons. If this is the case, then above the position of the unsplit peak the strength function for 3/2+ resonances will be greater than that for 5/2f resonances, as the spin-orbit term raises the peak for J = / - l/2 as compared to the one for J = I + l/2; below the peak the opposite would be true. Thus the observed effect is entirely to be expected if the unsplit position of the peak is at A = 46 or 47 instead of 50; since we are observing the strength function at -3 MeV proton energy, instead of at zero energy, this is quite reasonable. While the precision of the present data is insufficient for it to constitute a measurement of the degree of splitting of the peak in any quantitative sense, the order of magnitude of splitting indicated is a few units of mass number. The absolute magnitude is the least precisely measured aspect of the quantity S*. Even so, the high values of S* found in the compound nucleus 51Mn indicate that the S-wave as well as the D-wave strength function is well above the black nucleus value for this nucleus. At first sight this is a little difficult to understand on the assumption of an S-wave strength function peak at A g 70, as a low S(p’) at A = 51 would seem to be implied. However, a recent survey of S-wave neutron strength function data (46) indicates that for even-even targets the 3s peak is somewhat lower, much wider, and rather less clearly defined than the calculated curves usually seen. While no satisfactory explanation for this behavior has been proposed, there is no particular reason to believe that the 3s proton peak predicted at A = 70 should not be similarly distorted; and thus a high S( p’) at A = 51, even with a lower one indicated at A = 47, is not necessarily unreasonable. If this is the case, the low S* measured for 57Co would imply that the D-wave strength function is pretty well down from its peak at A = 57, a further indication that the D-wave peak may be somewhat lower than A = 50. From all this we see that, while the present data cannot yield direct strength function measurements, the qualitative inferences which can be drawn from the measured quantity S* are in good agreement with currently expected trends in the strength function. As has been mentioned, there is evidence in the present data

160

KYKER,

BILPUCH,

AND

NEWSON

for the existence of the D-wave strength function peak at A E 47 which can be adduced without reference to absolute yields. It is immediately obvious from the location of the experimental points in Figs. 13, 14, and 15 that a significantly greater number of the resonances which must be assigned 512% in the compound nuclei 47V and 51Mn require the assumption of large D-wave admixtures than is the case in 57Co. This indicates that TijD , the mean reduced width for D-wave proton emission, is larger relative to yijS for the lighter targets, and hence that the D-wave strength function is larger for them. This constitutes further evidence that there is a maximum in the D-wave strength function near or below A = 51.

ACKNOWLEDGMENTS We would like to express our indebtedness this experiment and provided valuable counsel Meier for their assistance in taking the data. RECEIVED:

to Professor R. M. Williamson, who suggested in its early stages; and to D. L. Sellin and M. M.

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