High-resolution proton scattering from 48Ti

11.E.I:Z.L]

Nuclear Physics A194 (1972) 353--379; (~) North-HollandPublishing Co., Amsterdam

Not to be reproduced by photoprint or microfilmwithout writtenpermissionfrom the publisher

H I G H - R E S O L U T I O N P R O T O N SCATTERING F R O M 4aTi N. H. PROCHNOW t, H. W. NEWSON and E. G. BILPUCH Duke University and Triangle Universities, Nuclear Laboratory, Durham, North Carolina 27706 tt

and G. E. MITCHELL North Carolina State University, Raleigh, North Carolina 27607 and TUNL, Durham, North Carolina 27706 *t

Received 8 May 1972

Abstract: Differential cross sections were measured for 48Ti(p, p) and 4aTi(p, Pl) at four angles between Ep = 1.8 and 3.1 MeV. The overall energy resolution was 250--350 eV. Spins, parities, total widths and partial widths were extracted for 301 resonances. Two analogue states were observed, and spectroscopic factors and Coulomb energies determined for these analogue states. A large positive correlation was observed between the elastic and inelastic widths for one analogue; no correlation was observed away from the analogue. The spacing distributions of the s~: and p~: resonances (after correction for the energy dependence of the average spacing) are in reasonable agreement with the Wigner distribution. The s~, p~_ and Pk proton strength functions were determined. E

NUCLEAR REACTIONS 4STi(p, p), 4aTi(p, Pl), E = 1.8--3.1 MeV; measured or(E, 0). 49V deduced isobaric analogue resonances, J, ~, _P,/~p, zrpl, Coulomb energy differences, nuclear temperature, level spacing, strength function.

1. Introduction This paper describes one of a series o f high-resolution m e a s u r e m e n t s of p r o t o n i n d u c e d reactions in the A ~ 40-64 mass region. These experiments have a dual purpose: to study the fine structure of analogue states 1 - 4 ) a n d to o b t a i n statistical i n f o r m a t i o n f r o m the "off-analogue" resonances. F o r the Ti isotopes the ( c o m p o u n d nuclear) level density is large e n o u g h that the parameters o f a large n u m b e r o f levels can be obtained, yet low e n o u g h that for some spins almost all of the levels are observed. I n this paper the results for ~ 300 levels observed i n 4 8 T i + p are presented. Earlier papers in this series covered analogue states from p r o t o n scattering 5 - a ) o n the even Ni, Fe a n d Cr isotopes, a n d n e u t r o n decay f r o m analogue states 9). A prel i m i n a r y report o n some o f the statistical results o n 48Ti has been published t o). F u t u r e p u b l i c a t i o n s will include, in addition to elastic scattering data o n other elements, results on the (p, Y) reaction t h r o u g h fragmented analogue states, studies o f * Now at University of Wisconsin River Falls, River Falls, Wisconsin. t* Work supported in part by the US Atomic Energy Commission. 353

354

N.H. PROCHNOW

e t al.

the fine-structure distributions, proton strength functions and more detailed statistical studies. Sect. 2 briefly describes the experimental procedure and the methods of analysis. The data and results of the resonance analysis are presented in sect. 3. In sect. 4 the analogue states are identified and spectroscopic factors and Coulomb energies are calculated for these analogues. Correlations between the elastic and inelastic widths are discussed in sect. 5. Statistical considerations are discussed in sect. 6, with an emphasis on the spacing distributions for the various species of levels. Average proton strength functions are determined for J~ = ~1+ , ½- a n d S - .

2. Experimental procedure and analysis 2.1. EXPERIMENTAL These experiments were performed using the high-resolution proton beam of the T U N L 3 MV Van de Graaff accelerator. The electrostatic analyser-homogenizer sys tern has been described in earlier papers in this series. Targets were prepared by evaporation of titanium (metal powder) onto carbon backings of thickness < 10 /ag/cm 2. The targets were 99.1% 48Ti of thickness ~ 1.5/~g/cm 2. The charged particles were detected using surface-barrier detectors. The semi-automated data-acquisition system 11) has been described previously 6). For the beam intensities used ( ~ 5 /IA), a total integrated charge of 150-300/aC maintained 1 % statistics in the offresonance cross section. The average counting time per data point was 30~,0 sec. The elastic data were taken at 160 °, 135 ° 105 ° and 90 °, while the inelastic data were taken at 160 °, 135 ° and 105 ° or 90 °. The measurements for elastic and inelastic scattering were made simultaneously. Near-resonance data were taken in energy steps of 100 eV, while off-resonance data were measured in energy steps < 400 eV. The resolution was maintained at 250-350 eV ( F W H M of Gaussian resolution function) throughout. Most of the more complex regions were taken with resolution __< 300 eV. Improved detector collimation reduced the background in the vicinity of the inelastic peak as compared with our earlier experiments. The improved inelastic data were critical in determining the spin of many of the resonances. Absolute energy calibration was achieved using the 7Li(p, n) threshold. Measurement of the analogue of the ground state of S°Ti(Ep ~ 1.37 MeV) yielded an absolute energy within 1 keV of that obtained by Maripuu 12). Although the absolute calibration is accurate only to several keV, the relative energies over a small energy range should be reproducible to within 300 eV. 2.2. ANALYSIS All of the data were analysed using a multMevel, multi-channel R-matrix computer program 13). The following resonance parameters were varied: spin J, parity re, resonance energy E o, elastic scattering width Fp, inelastic scattering widths Fr,~' (where 1' and s' are the orbital angular m o m e n t u m and the channel spin in the exit

4STi PROTON SCATTERING

355

channel) and the resolution width A. The capture channel was neglected and kinematic and penetrability considerations eliminated all particle exit channels except elastic and inelastic. The resolution was determined locally by fitting narrow resonances. Hard-sphere phase shifts (to l = 4) were included in the calculation- these hardsphere contributions to the cross section are not significant for most of the data, but do amount to about 13 ~ of the cross section for the worst case (160 ° and Ep = 3.1 MeV). TABLE 1 '~STi(p, Pl ) exit channels Initial state j~r

l'

s'

½+ ½~÷

2

~-,~

k~+

1

~,{ ~,~

0 2 l

~,~ ~,~

~-

l

0 2

Inelastic decay to the 2 + state; l' ~ 3 is neglected. Spin assignments were based on shape analysis of the elastic resonances and upon the inelastic angular distributions. Spin-½ + levels presented no difficulty. Spins p~ and p~ could be distinguished for resonances with Fp => 30 eV on the basis of elastic scattering alone. For resonances with inelastic decay, p~ and p~ resonances could normally be distinguished for elastic widths as small as Fp ~ 15 eV. The d-wave resonances were typically rather small (average laboratory width Fp ~ 10-15 eV) and as discussed below, it was not usually possible to distinguish between d~ and d~ resonances. For consistency, d-wave resonances of unknown spin were assumed to have J~ = ~+ in the analysis. A similar ambiguity also existed between f_~ and f~ resonances, but very few f-wave resonances were observed. The inelastic scattering proceeds to the 2 + first excited state of 48Ti at 0.983 MeV 14). The allowed orbital angular momenta and channel spins in the exit channel are listed in table 1. Sample calculated inelastic angular distributions are shown in fig. 1. Due to the very low penetrabilities, l' > 3 was neglected. The ½+ states decay with l' = 2 and s' = -~, ~. Since the inelastic angular distribution is isotropic, it is impossible to distinguish between the channel spins. For the few s-wave resonances with a measurable inelastic width, the widths for channel spin 8 and -~ were arbitrarily set equal. Assuming only l' = I, the ½- resonances have a unique channel spin (s' = 8) in the exit channel. The angular distribution is isotropic. For ~:- states, there is a mixture of channel spins (s' = 8, ~2). The inelastic widths for the two channel spins were adjusted to fit the observed angular dependence of the

356

N.H. PROCHNOWet al.

inelastic scattering. The d-wave resonances can decay by 1' = 0 or 2. Surprisingly the inelastic angular distributions from d-wave resonances were often anisotropic, indicating appreciable l' = 2 width. For decay o f ~2÷ (~+) resonances the 1' = 0 component has mixed channel spins. The following procedure was adopted in the analysis

|

I

I

,z[

I

I

'

I

j~=3/z+

I

I

I

q

'

I

'

I

J

I

j~ =5/2 +

5

\

E 0 .!

2l

Q-° ~ ; . ~

~o .~°,~

Q=2 s-5/2

rp, ,20eV

~=0 ~=5/2

p=-20 eV

Q=2 s'=5/2

rp=-ZOev

/ 0/=

I

mo

I

=

I

*

I

,zo =40 OLAs (deg)

'

I

'

I

I

I

~6o

=

I

,

I00

I

120

,

I

1

I

I

I

t

14o

~eo

140

160

OLAB (deg) '

I

'

I

,

I

'

JT=5/2-

I

d

20

,~-I s • 3~pl-

2°t

20eV

15

JQ E

Iol

~'~'5/z I

I I00

,

rPt=2oev I

120

=

I

t40

=

I

160

=

too

0LA s (deg)

Fig. 1. S a m p l e

calculated

mo

0LAB (deg) angular

distributions

for inelastically

scattered

protons.

4STi P R O T O N S C A T T E R I N G

357

o f d-wave resonances: A first fit was obtained assuming only l' = 0. I f the fit was not satisfactory (anisotropic angular distribution), an l ' = 2 width was added. The l' = 2, s = } and { widths were adjusted to yield the best fit (while reducing the l' = 0 width). As a result o f these complications in the analysis o f inelastic scattering for d-wave resonances, the inelastic scattering data was not usually o f much help in distinguishing between d,} and d}. 590

. . . .

48Ti(p,p)"sTi 2~

~'~-'~

~"~',~~-

i

SLy,B= 160" !J

"

i

~

~

~

'

-

'i ~'~" q-~""~". . . . . ""iTM

!OC

_ _

I

-

185

-

I

L

~90

195

2.~)0

!

2O(

,

205

I

ZlC

1

215

,

!

-Q

E i00

'I ot

0L

--Jr

I

-I

I

r

~

I

4"Ti (p,pl)4eTi t

2.20

I

225

250

2.55

Ep(MeV)

240

2.45

2.50

Fig. 2. The 4STi(p, p) differential cross section f r o m 1.80 to 2.50 MeV at 160 °. Although inelastic data were taken f r o m 1,80 MeV, no resonances were observed until 2.33 MeV. The 4aTi(p, Pl ) differential cross section is s h o w n f r o m 2.30 to 2.50 MeV. The solid line represents a fit to the data.

The inelastic data were quite valuable in distinguishing between p~ and p~ resonances and in obtaining better values for the elastic widths. The errors in the elastic widths range f r o m 10 % for very strong resonances to as m u c h as 50 % for very weak resonances. The larger inelastic widths should be accurate to 25 ~o or better. Reduced widths were obtained f r o m the laboratory elastic or inelastic widths using 72 = Fp/2Pc. Penetrabilities were calculated f r o m C o u l o m b wave functions evaluated at a channel radius ac = ao(A~+A~z) fm with ao = 1.25 fm. 3. Data Figs. 2 and 3 show the elastic and inelastic excitation functions at 160 ° lab angle. A l t h o u g h inelastic data were taken f r o m 1.80 MeV, the first resonance in the inelastic

358

N.H. PROCHNOW

et aL

cross section was not observed until 2.33 MeV. The solid curves t h r o u g h the data points in figs. 2-6 are fits to the data. The absolute cross sections are determined by normalising to Rutherford scattering. Fig. 4 presents an overall view of the data. The rapid increase with energy in the number o f observed resonances in both elastic

48Ti (p,p)48Ti

~9,A =160 °

200

I00

C

I

I

I

4~Ti (p.p~)4~Ti

2c

1

fC

2. _Q

c

255

30C

260

2.70

265

2 75

280

E 20C

t0C

--------+

+____

4

20

1 285

t 290

2.95

l 5.00

Ep(MeV)

5 05

5JO

Fig. 3. The 48Ti(p, p) and 48Ti(p, p~) differentia] cross section f r o m 2.50 to 3.10 M e V . The solid a r e located

line represents a fit to the data. The analogues of the 17th and 18th excited states of 49Ti near 2.85 and 2.95 MeV. They are assigned ½- and ~-, respectively.

and inelastic scattering is obvious - this is primarily a penetrability effect. The lower portions o f fig. 4 show the data and fit averaged over 5 and 20 keV, respectively, to simulate the results o f a poorer-resolution experiment. Figs. 5 and 6 show fits to the data on an expanded scale in the region o f the analogue state at Ep ~ 2.95 MeV. These results at four angles for the elastic scattering and three angles for the inelastic scattering indicate the quality of fit obtained. Table 2 lists the parameters o f the 301 resonances analysed.

*8Ti P R O T O N S C A T T E R I N G

359

4. Analogue states 4.1. G E N E R A L

Interpretation of the data is facilitated by plotting the cumulative sum of reduced widths for each spin and parity separately. Fig. 7 shows such a plot for the ~STi elastic data. Since the d~ and d~ resonances could not usually be distinguished, all

~o

4~Ti( p 'P~) 4STi'~

E)'AB= 160° t

2C

!

I

I

j

,,

i~

- .=. ~ d , . h,~

,

,

,

48Ti( p, p)*STi

20(

• -o~

~o0

~ 0

I

I

I

~

}

~O0

IOC ,v , ~E 0

- 5keY

2.

(AVG)

I

I I

~

I

-

20C ~

I

RUTHERFORD

-

....

RUTHERFORD -}- HARD SPHERE

•

~

,

~

IC)C AE -20keY (AVG) 01.80

2;0

220

2 . 4' 0

2 6I0

I 2.80

3 0I0

Ep (MeV) Fig. 4. The upper portion of this figure shows the 48Ti(p, p) and 4aTi(p, P l ) differential cross section ( ~ 300 eV resolution) at 160 ° over the entire energy range. The solid line represents a fit to the d a t a . The lower p o r t i o n of this figure shows the differential cross section averaged (both d a t a a n d fit) in 5 and 20 keV square intervals over the entire energy range. The figure also illustrates the hardsphere c o n t r i b u t i o n t o the cross s e c t i o n . A t 3.00 MeV the hard-sphere contribution to the cross section is about 10 % o f the total cross section.

d-wave widths are combined into one plot. Although the general trends are fairly smooth, there are a number of anomalies apparent in the data. The left-hand side of fig. 8 shows the results of a (d, p) experiment on 48Ti by Wilhjelm et al. 15) The (d, p) results suggest the following analogues might be ob-

360

N.H.

PROCI-INOW

e t al.

served: a weak l = 0 analogue and a strong l = 3 analogue near Ep ~ 2.2 MeV, and two l = 1 analogues at about Ep = 2.85 and 2.95 MeV. The reduced widths for the various spins and parities determined in this experiment are plotted for comparison

~Ti (p,p)4eT

50C

(~LAB = I 6 0 °

200 ,,

,

,~

~

V

I00

I

i

I

I

[

I

(~LAB = 155° 200

~/)100 ..Q

E

o

I

I

r

i

I

[

500

OLAB=I05

I °

'

200

1

I00

I

400

I

I

I

I

~.~2

' 2.9,

' 2.96 Ep (MeV)

' 2.~

I I OLAB= 90 °

300 200 IOC

~

~(~o

~.~,~

Fig. 5. The 48Ti(p, p) differential cross section at four angles, showing the elastic data on an expanded energy scale in the region of the ~- analogue state at 2.95 MeV. The solid line represents a fit t o t h e d a t a . T h e a n a l o g u e s t a t e at 2 . 9 5 M e V is d o m i n a t e d b y o n e v e r y l a r g e l e v e l at 2 . 9 5 3 M e V . T h e r e a r e t h r e e l a r g e s - w a v e r e s o n a n c e s l o c a t e d n e a r t h e a n a l o g u e state; t h e y are e a s i l y i d e n t i f i e d b y s t r o n g dips at 9 0 ° .

in fig. 8. The results s h o w in figs. 7 and 8 may be used directly to m a k e tentative identification o f the analogue states. N o f-wave analogue is observed, presumably due to the very low 1 = 3 penetrability. A positive identification o f the w e a k s-wave analogue near Ep = 2.2 M e V is not

'*STi PROTON SCATTERING

361

possible. There are a n u m b e r o f weak s-wave resonances in this vicinity, but there is no basis for a firm identification. The anomalies in the p{ and p{ strengths near Ep = 2.85 and 2.95 MeV are identified as the analogues o f the strong l = 1 states observed in the (d, p) experiment. The anomalously strong single levels at Ep ~ 2.15 M e V (p{), 2.70 MeV (s{) and 2.85 MeV (s}) apparently are n o t analogue states.

3°oF

4"Ti (p,p)48Ti

eLAB :160 o

20C ~'-

o

I

t

,..2o,- ,STi (p,p)4, Ti -~

Jo

o

I

I

.-

~

,'1-

j

OLAB : 16o"

"

I

k

-_.;,,.-

t t

~ L A B = 105°

20

I0

t 2 90

2.92

: 2,94

2 96

2.98

Ep (MeV)

' 3.00

3.02

Fig. 6. The *aTi(p, p) differential cross section at 160 ° and the 48Ti(p, p j ) differential cross

section at the three angles. This figure shows the inelastic data on an expanded energy scale in the region of the analogue state at 2.95 MeV. The solid line represents a fit to the data.

It is interesting to examine the overall behavior o f the inelastic scattering widths. On the average the s½ and p~ resonances did not decay inelastically, while the p~ and d-wave resonances usually had measurable inelastic widths. Since the s-wave and d-wave resonances decay with l' = 2 and I' = 0, respectively, the observed average behavior o f the inelastic widths for s-wave and d-wave resonances is expected f r o m penetrability considerations. It is easier to understand the relative strengths o f p~ and p½ inelastic widths in the j-representation, rather than in the channeI-spin rep-

362

N . H . PROCHNOW et al. TABLE 2 49V r e s o n a n c e parameters

Eo ~) (MeV)

j n b)

1.9019 1.9093 1.9240 1.9255 1.9787 1.9822 2.0128 2.0323 2.0336 2.0615 2.0777 2.0935 2.1014 2.1142 2.1163 2.1489 2.1496 2.1521 2.1563 2.1625 2.1702 2.1801 2.1805 2.1903 2.1936 2.2032 2.2045 2.2108 2.2121 2.2202 2.2239 2.2296 2.2308 2.2323 2.2543 2.2577 2.2638 2.2651 2.2661 2.2730 2.2806 2.2885 2.2912 2.2968 2.2981 2.3073 2.3136 2.3182 2.3190

(3-) (½-) ~(3-) (½-) ½+ ½+ (½-) ½+ ½+ ½(½-) (½-) (½-) (½-) ½+ ½(6 + ) ½+ ½÷ ~½+ (½-) ½(½-) (~+) ½+ (½-) (½-) ½+ ½+ (½-) (½-) (~+) ½+ (½-) ½(3 +) (½-) ½+ (.~-) ½+ ½(~-) (3 +) (~+) ½+ ½+ (;~+)

/~p (eV) 15± 5 15± 5 305- 10 205- 5 255- 5 15± 5 702:15 20:~ 5 7 0 i 15 705- 20 2605- 15 255- 5 205- 5 15± 5 55- 3 15___ 5 555- 10 5::k 3 105- 5 125J_ 25 40± 5 155- 5 205- 10 605- 10 15± 5 55- 3 25 5- 5 105- 5 205:5 7 0 ~ 20 190± 40 205- 5 25~ 5 55- 3 155- 5 20~ 5 9 0 ± 15 2-k 2 5± 3 20± 5 25± 5 405- 5 30± 5 25___ 5 105- 5 105- 5 15± 5 1 0 0 ± 15 55- 3

yp2 (keV) 1.08 1.05 1.99 1.32 1.36 0.33 1.40 0.90 1.31 1.20 10.10 0.92 0.72 0.52 0.17 0.20 1.71 0.79 0.13 1.59 1.16 0.18 0.55 1.65 0.41 0.67 0.28 0.26 0.5l 0.76 2.05 0.49 0.61 0.61 0.15 0.45 2.00 0.22 0.11 0.19 0.53 0.36 0.62 0.51 0.99 0.96 0.13 0.85 0.46

1' c)

s" ")

-Fpl (eV)

~pz (keV)

48Ti PROTON SCATTERING

363

TABLE 2 (continued)

Eo °)

j , b)

(MeV) 2.3204 2.3228 2.3340 2.3401 2.3464 2.3493 2.3550 2.3637 2.3644 2.3649 2.3709 2.3731 2.3751 2.3780 2.3890 2.3969 2.3977 2.4002 2.4008 2.4028 2.4034 2.4203 2.4248 2.4254 2.4263 2.4302 2.4376 2.4464 2.4477 2.4487 2.4567 2.4576 2.4624 2.4631 2.4660 2.4694 2.4776 2.4820 2.4850 2.4903 2.4949 2.4958 2.4988 2.5004 2.5035 2.5176 2.5207 2.5221 2.5344 2.5371

(~+) (½-) (~+) (½-) ½+ ½½+ (6 + ) (6 +) ½+ ½(½-) ½+ (½-) ½(½-) (6 + ) ½+ (½-) (~-) (6 +) (6 + ) ~½+ (6 +) (6 +) 4(4 + ) (½-) ½½+ (6 + ) .~½+ ½(6 + ) (4-) :~(½-) (6 ÷ ) (6 + ) ½+ ~+ ½½+ (~+) ½+ ½(½-) (~+)

F~

~,p~

(eV)

(keV)

lOi tO& 20± 15! 25 i 75~ 160± 10~ 7~ 230± 140& 5~ 90i 5~ 30± 20~ 20~ 20& 25~ lOi 5~ 5~ 501 165~ 3± lOi 30~ 5~ 25~ 50± 525~ 10± 30± ~± 225 ± 20± 20± 55± 15~ 5± 3± 130± 20± 30i 80± 15~ 10± 75~ 10± 10±

5 5 5 5 5 10 15 5 5 30 15 3 10 3 5 5 5 5 5 5 3 3 15 15 2 5 5 3 5 10 50 5 10 10 25 5 5 5 5 3 2 15 5 5 10 5 5 10 5 5

0.93 0.19 1.78 0.27 0.20 1.32 1.24 0.81 0.57 1.74 2.33 0.08 0.66 0.08 0.48 0.31 1.48 0.14 0.39 0.15 0.36 0.35 0.73 1.09 0.20 0.67 0.42 0.32 0.34 0.69 3.22 0.62 0.40 0.24 2.96 1.21 0.26 0.70 0.19 0.29 0.17 0.73 0.11 0.36 0.44 0.80 0.05 0.87 0.11 0.51

r c)

0

s' ~)

~-

F~

~2

(eV)

(keY)

2

0.98

0

2

0.49

1

10

6.54

I

5

3.01

0

2

0.43

N. H. PROCHNOW et aL

364

TABLE 2 (continued) Eo a) (MeV)

j n b)

/,p (eV)

7,pz (keY)

2.5380 2.5419 2.5468 2.5496 2.5547 2.5575 2.5657 2.5713 2.5734 2.5763 2.5773 2.5870 2.5872 2.5943 2.5962 2.5985 2.6008 2.6081 2.6107 2.6180 2.6187 2.6279 2.6313 2.6358 2.6371 2.6437

.~½+ (~+) ½+ (~+) ½+ (~-) (~+) ½½÷ ½(l] +) ½(1)+) (2 +) ½+ ½(½-) ]½+ (~+) (6 ÷) (~+) ~½(6 +)

3 0 i 15 10± 5 35- 2 1355- 15 55- 3 155- 5 155:5 35:2 1655- 15 275:~ 20 1005:15 55:3 3005:30 125- 5 5± 3 2505- 25 1855- 20 105: 5 705- 20 3005:60 15± 10 305- 10 125:5 1005:15 255- 5 35~_ 5

0.33 0.05 0.15 0.68 0.24 0.07 0.16 0.14 1.70 1.32 1.02 0.22 2.99 0.52 0.22 1.15 1.79 0.09 0.66 1.32 0.62 1.21 0.48 0.90 0.22 1.35

2.6480 2.6499 2.6525 2.6545 2.6560 2.6615 2.6641 2.6756 2.6764 2.6824

½+ (2 ÷) ½½÷ (~+) ½+ (½-) ½+ (~+) ~)+

2105- 15 10± 5 754- 20 3005- 30 15~- 5 5255- 75 154- 10 1605- 25 104- 5 655:15

0.88 0.38 0.65 1.24 0.56 2.13 0.13 0.63 0.36 2.29

2.6843 2.6857 2.6885 2.6915 2.6964 2.6983 2.6996 2.7007 2.7035 2.7051 2.7056

½+ (6 +) ~+ ½]~2+ ½÷ ½½~+ ~--

1904- 20 2± 2 5 5 i 10 2905:30 50± 5 65- 3 2205- 20 305:15 355- 10 35~ 10 1505- 25

0.74 0.06 1.91 2.32 0.40 0.20 0.83 0.24 0.27 1.17 1.17

l' ¢)

s ' ~)

/~pl (eV)

y2p~ (keY)

0

~

5

0.62

0 1 0

~ ~ ~

5 5 5

0.58 1.52 0.55

1

~

3

0.81

0 0

~ 2-

2 3

0.20 0.28

1 1 0 2

~ ~ ~ 2

3 9 3 1

0.71 2.12 0.26 1.39

0 2 2

~ ~ ~

8 10 10

0.59 11.43 11.43

0

~

5

0.36

0

~-

12

0.82

0 1

~ ~

5 10

0.33 1.70

z

Z 0 0

e q

F-

_

w

+

l

l

+

l

+

l

+

++

I

+

+

+

+

l

l

+

+

v

.

t

+

+

l

+

+

l

+

+

+

+ + + 1

+

366

N. H. P R O C H N O W et aL TABLE 2 (continued)

Eo ") (MeV)

s - ~)

F. (eV)

7~ ~ (keV)

2.8175 2.8225 2.8259

~~+ ~-

2.8288

(~+)

2.8317 2.8322 2.8330

½+ (~+) :~-

190~ 50 15± 5 3 0 ± 10

0.57 0.38 0.18

2.8333 2.8355 2.8369 2.8381 2.8390 2.8416

(~+) ½½½+ (~-+) 2-

2+_ 70~ 50± 50~ 5i llO!

2 20 10 10 3 15

0.05 0.42 0.30 0.15 0.12 0.66

2.8422

(~-+)

20±

5

0.49

2.8468 2.8514 2.8539

{{~J-

7 5 0 ± 75 2 5 0 ± 25 2 0 ± 10

4.46 1.47 0.12

2.8553 2.8610

½(,~+)

1500±200 4 0 ± 20

8.78 0.94

2.8611 2.8671

½+ (~+)

5 2 5 ± 75 154- 5

1.52 0.35

2.8679 2.8680

½(~+)

900 ± 150 4 5 ± 10

5.15 1.05

2.8687

3+

3 0 ± 10

0.70

2.8737 2.8790 2.8859 2.8913 2.8938 2.8964 2.8980 2.8992 2.9007

(~+) ½+ ~+) ½+ (½-) (~+) ½+ (½-) ~-

5± 3 2350 ± 200 2± 2 5 0 ± 10 10± 5 24- 2 5 0 ± 15 10± 5 4 0 ± 10

0.11 6.60 0.04 0.14 0.05 0.04 0.14 0.05 0.22

3 5 ~ 15 10~ 5 25~ 5 5~

3

0.22 0.26 0.15 0.13

r ~)

s ' °)

/'., (eV)

7., ~ (keV)

0 1 1 0 2 2

.~ ~ ~, ~ ~~

14 1 6 2 2 4

0.57 O.lO 0.61 0.08 1.15 2.30

0 I 1 0

~ :~ ~~

5 6 6 5

0.2O 0.59 0.59 0.20

1 1 0 2 2 1

~ ~~ :] ~ .~

4 5 2 1 1 20

0.38 0.48 0.07 0.54 0.54 1.86

1 1 1 0 2 2

] ~ ~ ~ ~ ~

5 1 40 1 1 2

0.45 0.09 3.60 0.03 0.50 1.00

0 2 2

~ ~ ~

2 4 4

0.06 1.95 1.95

0 2 2 0 2 2 0

~ .~ .~ ~ -:] ~ ~

15 2 2 2 3 6 5

0.51 0.97 0.97 0.O6 •.45 2.90 0.17

0

~

2

0.06

1 1

3 ~

5 2

0.38 0.15

367

4STi PROTON SCATTERING TABLE 2 (continued) Eo ") (MeV)

j n b)

I~ (eV)

2.9061 2.9082 2.9107 2.9147

.~+ (~+) (½-) ~-

140~ 3zk 15~ 100-5z

15 3 5 15

0.38 0.06 0.07 0.53

2.9186 2.9216 2.9227 2.9244

(,~+) (~+) (~+) (~+)

15Jz 8:k 5zk 30~

5 4 3 5

0.32 0.17 0.17 0.62

2.9284

(~÷)

18~ 5

0.37

2.9292 2.9315 2.9336 2.9378 2.9393

(~+) ½+ ½(~+) ~-

5:L 3 150:k 20 20~ 5 8-~ 4 85~ 15

0.10 0.39 0.10 0.16 0.43

2.9412 2.9443

~z+ ~-

285=k 30 300± 30

0.73 1.50

2.9453

]-

2 7 5 ! 30

1.37

2.9505

(~+)

2.9533

~--

2.9557

(~+)

2.9578 2.9602 2.9622 2.9651

½+ ½+ (,~+) ~-

2.9661

(~+)

2.9679

10±

~,p2 (keY)

5

0.20

2800 -k 300

13.81

10±

5

0.19

400± 50 325± 35 8+ 4 250± 30

1.00 0.81 0.15 1.21

22+

5

0.42

~-

70+ 15

0.34

2.9687

(~+)

4 0 ± 10

0.76

2.9716 2.9736 2.9740 2.9780

(25+) ](½-) (;~+)

5~ 3 4 0 ± 10 15:k 5 5:k 3

0.09 0.19 0.07 0.09

l' ¢)

s' ~)

/'p~ (eV)

~,zp, (keV)

0

~)

2

0.05

1 1 0 0 0 0 2 2 0 2 2 0

-.~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~

12 1 5 2 1 8 5 2 2 3 2 5

0.86 0.07 0.14 0.05 0.03 0.22 1.92 0.77 0.05 1.13 0.75 0.14

1 0 1 1

~ ~ ~ ~

2 7 4 4

0.13 0.19 0.26 0.26

1 1 1 1 0 2 1 1 0 2

~ ~ ~ ~ ~ ~ ~ ~ ~~

20 45 25 5 1 1 60 210 2 4

1.28 2.88 1.59 0.32 0.02 0.35 3.72 13.00 0.05 1.35

o 1 1 0 2 2 1 1 0 2 2 0 1

~ ~"i ~ ~~ .~ ~ ~ ~ ~ ~ ~z

1 2 13 6 2 8 4 18 15 2 5 3 25

o,o2 0.12 0.77 0.15 0.65 2.60 0.24 1.06 0.36 0.64 1.61 0.07 1.44

0 2 2

~ ~ ~

8 4 6

0.19 1.24 1.86

368

N. H. PROCHNOW et aL TABLE 2 (continued) 7~ ~ (keY)

I" ")

s' ~)

5

0.18

204254343541251254255-

10 10 3 5 5 25 10

0.37 0.12 0.01 0.16 0.22 0.30 0.45

0 2 2 0

(~+)

225-

5

0.39

3.0012

(~+)

204-

5

0.35

3.0066 3.0082 3.0145 3.0184 3.0215 3.0234 3.0256

½½+ ~½+ (~+) (~+) (~+)

75+ 225::t:: 75± 250412-55205-

15 25 15 50 5 3 10

0.34 0.52 0.33 0.57 0.20 0.08 0.34

3.0262 3.0289 3.0295

½+ (~+) ~-

9505-150 285- 5 30± 10

2.15 0.47 0.13

3.0300

(~-+)

3.0306 3.0311 3.0367 3.0375 3.0380 3.0383 3.0391 3.0435

~(~+) (~+) (~+) ](~+) ½½+

3.0442

(8 +)

3.0454 3.0485 3.0497 3.0527

(z+)2 (~÷) k(8+)

Eo a) (MeV)

S . ~)

I$ (eV)

2.9796

({+)

105-

2.9825 2.9849 2.9887 2.9902 2.9930 2.9949 2.9950

(~+) (½-) (~-) ½(~+) ½+ (~+)

2.9981

205-

5

0.34

15± 5 3i 3 85- 4 44- 3 255- 5 12± 5 100± 15 290± 60

0.06 0.05 0.13 0.06 0.11 0.20 0.43 0.64

12±

5

0.20

154- 5 55- 3 225 ± 20 225- 5

0.24 0.08 0.95 0.35

F., (eV)

7~, ~ (keY)

~ ~ ~ ~

2 3 4 2

0.04 0.92 1.23 0.04

1 0

~ ~

7 12

0.38 0.27

0 2 2 0 2 2 0 2 2

~ ]~ ~~ ~ .~ .~.~

18 2 2 4 3 2 32 2 2

0.40 0.58 0.58 0.08 0.86 0.57 0,64 0.57 0.57

0 0 0 2 2

~ ~ ~~ ~

12 3 4 2 2

0.24 0.06 0.07 0.52 0.52

0 1 1 0 2 2 1 0 0 0 I 0

~ 8~ ~.~ ~, -~,~~ ~ ~~

8 25 30 40 2 2 20 3 2 4 50 5

0.16 1.18 1.42 0.78 0.51 0.51 0.94 0.05 0.03 0.07 2.30 0.09

2 2 0 2 2 0

~] ~ ~ ,~ ~~

5 5 12 1 1 I

1.21 1.21 0.23 0.24 0.24 0.01

0

~

10

0.18

369

48Ti PROTON SCATTERING TABLE 2 (continued) Eo ~) ( M eV)

j~r b)

Fp (eV)

7p 2 (keV)

3.0559 3.0589

(:)+) ½+

8± 4 650± 100

0.13 1.40

3.0605 3.0628 3.0633

(;)+) (~2+) :~-

12± 5 2± 2 15± 10

0.19 0.03 0.06

3.0637 3.0674 3.0720

½½3-

2 5 ± 10 55 q- 20 20± 5

0.10 0.23 0.08

3.0752 3.0754 3.0783 3.0784

½+ (6 ÷) ½+ (~+)

40+ 10 5± 3 325± 75 30± 10

0.08 0.07 0.68 0.46

3.0800 3.0815 3.0843

½(~-÷) (~+)

2 0 ! 10 8± 5 104- 5

0.08 0.12 0.15

3.0850 3.0871

½+ ~.-

850 ±200 60q- 20

1.77 0.24

3.0878 3.0880 3.0905

½(~+) ~-

304- 15 4 5 ± 10 15± 5

0.12 0.67 0.06

3.1020

½+

lOOq- 25

0.20

l' c)

s' c)

/~pl (eV)

7P, 2 (keV)

0 2 2 0

~ :~ ~) .~

15 20 20 2

0.27 4.56 4.56 0.03

1 1 1 I

.~ ~ ~~

I !

~ ~

5 10 65 140 10

0.21 0.42 2.74 5.83 0.41

10

0.41

0

~

5

0.08

0 2 2

~ :) ~

8 5 5

0.14 1.06 1.06

1

~

25

1.00

0

~

12

0.20 0.08 0.42 0.42

0

~-

5

2 2

~ ~

2 2

1 1 1 0 1

~ ~, .~ ~ ]-

20 10 10 45 3

1

~

3

0.78 0.39 0.39 0.73 0.12 0.12

a) Laboratory energies are quoted. Although the absolute energies are accurate only to ~ 3 keV, the relative energies over a small energy range should be reliable to ~ 300 eV. b) Doubtful spin assignments are listed in parenthesis. For d-wave resonances for which not even a most probable assignment could be made, a spin of ~-+ was arbitrarily assigned. ~) Inelastic parameters are tabulated according to exit-channel orbital angular momentum 1' and exit-channel spin s'. Analysis procedure is discussed in sect. 2.

r e s e n t a t i o n . A ½- r e s o n a n c e c a n d e c a y (by I' -- 1) t o a 2 + state o n l y b y j p , = ~, w h i l e a ] - r e s o n a n c e c a n d e c a y t o a 2 + state byjp, = ½ o r s). T h e p~ s t r e n g t h f u n c t i o n in this m a s s r e g i o n is m u c h larger~ t h a n t h e p~_ s t r e n g t h f u n c t i o n . T h i s is o b v i o u s f o r 4ST± in t h e p l o t o f r e d u c e d w i d t h s s h o w n in fig. 7, a n d is c o n s i s t e n t w i t h o u r o v e r a l l results o n p r o t o n s t r e n g t h f u n c t i o n s 16), w h i c h i n d i c a t e t h a t t h e 2p~ g i a n t r e s o n a n c e is n e a r A = 40 a n d t h a t t h e 2p~ g i a n t r e s o n a n c e is b e l o w A = 40. T h u s t h e relatively s t r o n g inelastic d e c a y f r o m p~ r e s o n a n c e s f o l l o w s f r o m s t r e n g t h - f u n c t i o n c o n s i d e r a tions.

370

N.H. PROCHNOW

et al.

The left-hand side of fig. 9 shows the behavior of the inelastic reduced widths for ~ - resonances. Both channel spins show an anomaly at the a2- analogue state near 2.95 MeV. The right-hand side of fig. 9 shows the relatively smooth behavior of the inelastic reduced widths for d-wave resonances. 48Ti(p, p) 48Ti 40 I 12 RESONANCES

f f

/

20 0 60

I

'

'

'

I

'

. . . .

j~r = I/270 RESONANCES

Ij~2-

#

40

J

J

2O v

~

j

b,l::[ 0

0 8d

F I

.

.

.

.

I

40

RESONANCES

I

'

79

'

'

'

.

.

.

.

I

t

[

'

'

'

'

1

RESONANCES

60

_/7

40 20 f

2.o

2!s

Ep(MeV)

3!o

Fig. 7. C u m u l a t i v e s u m of reduced widths from 48Ti(p, p). The large anomalies in the 2 - and :~d a t a at Ep = 2.85 a n d 2.95 M e V indicate analogue states.

4.2. S P E C T R O S C O P I C

FACTORS

The spectroscopic factor for an analogue state may be calculated from Spp = (2To + 1)rp;rs.,.,

4aTi PROTON

SCATTERING

371

where To is the isospin of the target, Fpp is the total observed width of the analogue state (obtained by summing over the fine-structure fragments) and Fs.p. is the proton single-particle width. The single-particle width was calculated with a computer program written by Harney 17); a discussion of the methods of calculation is given by 49Ti LEVELS 3.50-

48Ti(p,p)aeTi LEVELS ~'2-(79 LEVELS) 3/~(40LEVELS) I/2"(70 LEVELS) A L L

LEVELS

(50i

L=I l=l

F

F 325--- -

-5,00

t,~ 2-1

-2.75 o-

z-

300-

L-

-2.50

E

~<

L

2;75-

U.I

LcJ

)

-2.25 r

250.J~_._ 11 =-0 3 F-=.

-2.00

225. _ . . . .

o i (2J +I)S

g

,b %

~

,o' o't

keV)

~

,'o

~

,b

F i g . 8. Summary of the present results on 4 a T i ( p , p ) and the 4 S T i ( d , p ) 4 9 T i data of Wilhjelm et al.

Harney and Weidemiiller l s). The procedure followed is discussed in preceding papers in this series. The ½- (4 fragments) and ~z- (9 fragments) analogues are well above the background and there is little doubt concerning which states to include as part of the analogue. The T< background strength is estimated (from average reduced widths in the vicinity) to be no more than 10 % of the analogue strength. The results are listed in table 3. The spectroscopic factors Spp are 0.13 and 0.12 for the k- and ~ - analogues, respectively, as compared with Sdp = 0.16 measured

372

N.H.

et al.

PROCHNOW

for both states in the (d, p) experiment. Considering all of the uncertainties involved this should probably be considered good agreement. 50~

~'ol

4BTi (p.p) 4a T, ,I ~=

......"

(3/2".5/2 +) rl2 RESONANCES

f ,.,,1

4aTi (p,p)AaTi jw= 31~

40 RESONANCES

f

3ot

i

I

c

30

2C

i

i

zot

[

Eot

J

'J

i

ol

fJ

4eTi( P.P,)*BTi

j~ ~F~ A

.>.

0

F

,

1"-

CHANNELSPIN=3~

i,,l::L IO

/

q 3O

r

j-

4°T i ( p. p,)*eTi fl

,<

O CHANNELSPIN:5/2.3~

S ~

/

rJ

Z'=2 CHANNELSPIN=3/2 /

Fr ~1

CHANNEL SPIN =5/2

J jI

,~'=2 CHANNEL SPIN ,5/2

20

F;

J ,["

z.6o

I

z[o Ep(MeV)

J

S

3bo

I zoo

zgo

Ep(MeV)

3oo

Fig. 9. (a) Cumulative sum of reduced widths for the ~ - states observed in the elastic and inelastic data. The large anomalies near 2.95 M e V indicate an analogue state. (b) Cumulative sum of reduced widths for the d-wave resonances observed in the elastic and inelastic data.

4.3. C O U L O M B

ENERGIES

The Coulomb energy difference between the parent and analogue states is given by

AEc =

B n + E , cp. m . - E x ,

where B, is the binding energy of the last neutron in the parent system and E x is the excitation energy of the parent state in the parent nucleus. The energy of the analogue state is determined by fitting 19) the fine-structure distributions to a Robson distribution l) or, if the state is not highly fragmented, by evaluating the centroid. The

4aTi PROTON SCATTERING

373

centroid was determined for the ½- analogue, while a fit to a Robson distribution was performed to the ~t- analogue. The neutron binding energy and the parent-state excitation energies (and uncertainties) were taken from Nuclear Data 2o). TABLE 3 Properties of observed analogue states ½- analogue Ecx of parent (MeV) Ep lab (MeV) Ep.... (MeV) /'pp (keV) /~p (keV) Spp

~- analogue

3.176 (8) a) 2.856 2.798 (3) 3.40 130 0.13 0.16 b)

Sap Bn (MeV) d e c (MeV) AEc others (MeV)

3.261 (8) a) 2.951 2.892 (3) 3.95 160 0.12 0.16 b) 8.143 (7)a)

7.765 (10)

7.774 (10) 7.822 (4) c) 7.730 (30) d) 7.807 (18)c)

All errors in keV in parentheses. a) Ref. zo). b) Ref. is).

~) Ref. 22).

a) Ref. 2a).

e) Ref. 24).

The Coulomb energies are listed in table 3. They are in excellent agreement with values calculated from Janecke's semi-empirical formulae 2~) and in reasonable agreement with previous experimental values 22-24). 5. Correlations between elastic and inelastic widths

Correlations are expected between partial widths of the fragmented analogue states for some of the decay channels 4). In simplest terms, correlations are expected between those channels which are enhanced by the analogue state. The g2- analogue state at Ep ~ 2.95 MeV provides an example where both elastic and inelastic widths are enhanced (see fig. 9). Fig. 10 shows the p~ elastic and inelastic reduced widths in the vicinity of this analogue. One quantitative measure of the correlation between two sets of numbers is the linear correlation coefficient r~___

N Z (xl Yi)-( ~,, xi)( E Y,) [N Z x ~ - ( 2 x,)2]~[ N Z Y~- ( Z y,)Z]~ '

where N is the number of pairs of numbers (xl, Yi). For the nine resonances which constitute the analogue, the linear correlation between the elastic widths and the channel spin ~ and ~ inelastic widths is 0.98 and 0.88, respectively. These results are

374

N . H . P R O C H N O W et al.

significant at a confidence level o f > 99 ~ and > 98 ~ , respectively. Since the finestructure distribution is dominated by a single strong level, the correlations are also dominated by the strong level. Even without the central strong level, the correlations are still o f order ~ 0.6, with a confidence level o f > 90 o/ W e have also observed a very strong correlation between elastic and inelastic widths for an analogue state with m o r e fragments and with a m o r e Lorentzian fine-structure distribution a s). /O

•

48Ti(pop)48

12

j'rr=

hi

Ti

5/2-

"

0

"--~

Ill

12

I

'

'

h

I

I

48 Ti (P'Pa)4 8 Tii"

OJ CHANNEL SPIN " 5/2 R A • 0.98 8

ROA.

0.03

! CHANNEL SPIN " 3 / 2

R, •

o.oa

Ro~ • - O. ~7

II,I

I 2.80

2.95

2.90

E

,,J

3~o

3.~0

I

I,

(MeV)

Fig. 10. Distribution of reduced widths near Ep = 2.95 MeV for ~-- resonances in 48Ti(p, p) and 4STi(p, Pl ). Here RA is the linear correlation coefficient between elastic and inelastic widths in the vicinity of the analogue, while RoA is the coefficient for states away from the analogue.

It is also interesting to evaluate the correlation o f elastic and inelastic widths away from the analogue. A w a y f r o m the analogue a correlation o f 0.03 was measured between elastic widths and channel spin ) inelastic widths (for -~- resonances). A small value is expected for ordinary compound-nuclear ( T < ) states. This result also tends to indicate that the correlations observed near the analogue state are real, and

48Ti PROTON SCATTERING

375

are not introduced by some systematic bias in the determination of elastic and inelastic widths.

6. Statistical properties 6.1. GENERAL There are sufficiently many resonances analysed in these 4 8 T i + p data to permit consideration of at least some of the statistical properties of these levels. Such properties as densities, spacing and width distributions, and strength functions have been extensively studied via neutron reactions 2 6). Usually these neutron studies are limited to s-wave resonances and to a rather narrow energy range. A recent summary of the status of nuclear statistical properties is given in ref. 27). The relative advantages and disadvantages of the proton and neutron resonance reactions should be made explicit: proton scattering provides results on s-, p- and d-wave resonances. A large energy range is covered conveniently, since the resolution is a constant as a function of energy. One basic limitation is that due to fixed energy resolution, increase in level density and strong penetrability effects, the proton measurements are at present limited to mass number A ~ 65. The number of resonances of a given spin and parity observed in a particular isotope is also rather limited. An examination of the densities and of the spacing and width distributions also provide an excellent opportunity to examine our data for internal consistency, and to help assess the correctness of spin assignments and the accuracy of width determinations. The number of levels of each spin-parity observed is 70(s~), 79(p½), 40(p~) and 112(d~, ~). If all spin-½ resonances are observed, the expected number of p~ resonances and d~, ~ resonances is ,~ 120 and ~ 260, respectively. (This assumes p(J) ~ ( 2 J + 1) e - ( s + ~)/2~2, and a spin cut-off factor a = 2 estimated from Gilbert and Cameron 28).) The number of observed s~ and p~ levels is about equal which is consistent with observing almost all of the spin-½ levels. Many d-wave resonances are not observed, due primarily to the low d-wave penetrability. More surprising is the very large number of p~ resonances missed. As discussed in subsect. 4.1, the p~ strength function is rather low in this mass region. The smallness of the p~ strength function, as well as the density effect (for the same strength function, the average reduced width for p~ resonance is about half that for p~ resonances) may explain the large number of p~ resonances missed. The levels are missed selectively: as is shown in fig. 7, only four p~ resonances were observed below ~, 2.4 MeV. Thus only the s~ and p~ resonances appear suitable for statistical analysis. For the s-wave resonances in 49V, most of the levels are observed, there are no ambiguities in the spin assignment and the effect of analogue states is minimal. Therefore, the analysis of these 70 s-wave resonances was emphasized in ref. lo). For present purposes we limit the discussion to rather qualitative aspects of the statistical properties

376

N . H . P R O C H N O W et al.

o f these data. More detailed statistical considerations of the entire body of highresolution data are in progress. .

x

.

.

.

60

~

,-

60

(o)

z re"

40

~

40

~ 20

.

20

w IE °

+

bYL\

z

•

1.0

0.0

2.0

I

3.0

i

i

4.0

I/ 0

0.0

1,0

2.0

i

!

3.0

4.0

x'= s~

X = S/(D) i

50

X 0 I-Z

z

120

(d)

u~ 9o o. z ~

60

~

ao

<> 5

z i

I 2.0

I 2.5 Ep

I 3.0

(MeV)

\ i

0 0.0

LO

j I-----r-'--V~ 2.0

3.0

4.0

X = S/
Fig. 11. (a) Uncorrected spacing distribution for the 49V ] - levels. (b) Corrected spacing distribution for the +gv ½- levels. (c) Local average level spacing for d-wave resonances in +9V. (d) Uncorrected distribution of the +gv d-wave spacings.

6.2. S P A C I N G A N D W I D T H D I S T R I B U T I O N S

The observed level density for the s-wave resonances in +9V changes over the energy range covered. This change is ascribed to a real change in the nuclear level density. The results are presented in a previous publication 1o). In the simplest model 26) the spacing may be taken to have an exponential dependence on the excitation energy, ( D ) ~ exp (-Eex~/T), where ( D ) is the local average level spacing, Eexc is the excitation energy and T is the nuclear temperature. In practice it is rather difficult to extract a precise nuclear temperature from the present data (due to the poor statistics).

48Ti PROTON SCATTERING

377

A value T ~ 1.5 MeV seems reasonable 28) and was adopted. The energy dependence of the local level spacing is consistent with this nuclear temperature. The distribution of spacings (for a given spin and parity) may be characterized by the Wigner distribution

P(x) = 5=x e -i~x*, where x = s / ( D ) , s is the spacing and ( D ) the average spacing. Since there is only average spacing in the Wigner distribution, and the data show a variable spacing, a "correction" must be applied to the data in order to make a direct comparison between the experimental and Wigner spacing distributions. The procedure is described in ref. 10). The corrected spacing distribution for the s-wave resonance is in much better agreement with the Wigner distribution. The statistical properties of the 79 5 - states in 49V should be qualitatively similar to those for the 70 s-wave resonances. The observed local average level spacing exhibits an energy dependence; the value T = 1.5 MeV is again consistent with the data. Figs. 1 l a and b show the uncorrected and corrected spacing distributions with normalised Wigner distributions superimposed. The observed corrected spacing distribution qualitatively agrees with the Wigner distribution, with poorest agreement near x = 0. The Wigner distribution has a value of zero at x = 0 (the level-level repulsion property), while the spacing distribution for resonances of different J~ does not. The observed spacing distribution near x = 0 for the 5 - resonances in 49V resembles the spacing distribution for two superimposed Wigner distributions 26). Since some ambiguity exists in the spin assignment for weak 5 - resonances (see subsect. 2.2), the discrepancy between the observed spacing distribution and the normalised Wigner distribution near x = 0 is attributed to this ambiguity in spin assignment. In fig. 1 lb the first bar of the data histogram corresponds to 11 spacings. The Wigner distribution for one J~ predicts that there should be ~ 4 spacings for this interval. Thus, if the poor agreement near x = 0 is attributed to ambiguities in the spin assignment, the number of levels misassigned is ~ 7. There are 18 5 - levels which have an elastic laboratory width of 15 eV or less and no inelastic width. These spin assignments are doubtful. Thus the behavior of the 5 - spacing distribution near x = 0 is probably due to incorrect spirt assignments for a few weak levels. F o r the J > 5 states a significant number of levels are not observed. The results of attempted analysis of these data are illustrated by the results for the 112 a2 +, -}+ states. Fig. 1 lc shows the observed local average level spacing for the 3+ states in 49V 2 , ~+ 2 The spacing does exhibit an exponential energy dependence, but the energy dependence is much too steep. The slope yields a nuclear temperature T ~ 0.5 MeV, 3 times lower than the value used for the s-wave and p-wave resonances. This behavior for the local average spacing is expected if a large number of resonances are missed in a selective manner. It is clear from the data (see top part of fig. 7) that many more d-wave resonances are being missed at lower bombarding energies than at higher energies. The average

378

N . H . P R O C H N O W et al.

reduced width for all observed d-wave resonances ( ( y z ) ~ 0.5 keV) corresponds to 2 eV at Ep = 2 MeV and Fp ~ 25 eV at E~ = 3 MeV. The level ofobservability is a few eV for favourable cases and at least 5 eV for unfavourable cases (no inelastic scattering, many resonances interfering). Thus the number of d-resonances observed is dominated by the penetrability and the levels are missed selectively (many more at lower energies). Under these conditions the spacing distribution should deviate from a Wigner distribution and should be skewed towards small values of x. Fig. 1 ld shows that this expected behavior is observed experimentally. /'p ~

TABLE 4

Proton strength functions for 4 S T i ÷ p jn

½+ 3-

No. of levels

Ei (MeV)

Ef (MeV)

Z~(Tp2)~ (keV)

SF

66 53 20

2.•49 2.093 2.438

3.102 2.786 2.854

66,91 41.87 8.69

0.070 0.060 0.021

E r r o r a)

~0.013 4-0.012 ±0.006

") Statistical error only. The fractional statistical error is V2/n, assuming a Porter-Thomas distribution.

As discussed in refl 1o) the proton reduced-width distribution for s~ resonances agrees reasonably well with the Porter-Thomas distribution. Further comparison of the width distribution with a Porter-Thomas distribution indicates that perhaps 10 s~ levels are not observed (compared to 70 observed). Analysis of the width distribution of the p~ resonances is complicated by the presence of the analogue state and by the apparent inclusion of several p~ resonances. Although these data may well form a more complete set of states than previously measured by proton resonance reactions, the ambiguities and missed levels do make a thorough statistical analysis difficult. Further studies are in progress, both experimentally (to make the data set more complete and unambiguous) and theoretically (to perform more detailed statistical analysis). 6.4. S T R E N G T H F U N C T I O N S

Proton strength functions were also extracted from these data. Table 4 lists the proton strength functions obtained for ½+, ½- and ~ - resonances (d-waves are not included since they represent an unknown admixture of d.~ and d r resonances). The strength functions were calculated from SF = Z

(~'2)I/AE,

i

where the sum includes all T< levels observed in the interval AE. The intervals were chosen to exclude the analogue states, as well as regions where many levels were clearly being missed. Even if a number of levels are missed, the missing levels are the weaker

*8Ti PROTON SCATTERING

379

ones, and thus do not very strongly affect the strength function. The s~ and P~r strength functions are fairly large, while the p~ strength function is small. These results are consistent with the overall trend of the proton strength functions in this mass region [ref. 16)]. The authors would like to acknowledge Dr. A. M. Lane for helpful correspondence and discussions, Profs. L. C. Biedenharn and R. Y. Cusson fordiscussions on statistical aspects of these data, and the other members of our high-resolution group for advice and assistance in performing both experiments and calculations. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 2l) 22) 23) 24) 25) 26) 27) 28)

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