Determination of the level width and density of 32S between 17 and 21 MeV excitation energy

2.D

[ I

Nuclear Physics A122 (1968) 4 6 5 4 8 0 ; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

DETERMINATION BETWEEN

OF THE LEVEL WIDTH

AND DENSITY

17 A N D 21 M e V E X C I T A T I O N

O F 32S

ENERGY t

H. K. VONACH Physik Department, Technische Hochschule Miinchen, Miinchen, Germany and Argonne National Laboratory, Argonne, Illinois 60439 and A. A. KATSANOS University of Washington, Seattle Washington 98105 and Argonne National Laboratory, Argonne, Illinois 60439 and J. R. H U I Z E N G A University of Rochester, Rochester, New York 14627 and Argonne National Laboratory, Argonne, Illinois 60439 Received 9 September 1968 Abstract: Excitation functions of the reactions alp(p, e0) and sap(p, ~ ) were measured with high resolution ( < 5 keV) at 7 angles in 10 keV steps for bombarding energies of 8.37 to 9.00 and 10.00 to 11.77 MeV. Auto-correlation functions were calculated from the excitation functions. From these auto-correlation functions, average widths of the compound states of a2S were determined to be 38±5, 4 7 ± 4 and 4 5 ± 5 keV for excitation energies of 18.1, 19.8 and 20.7 MeV, respectively. These widths are compared with statistical model calculations in which level densities of the Fermi-gas type are used. Good agreement between experiment and theory is obtained for level-density parameters which give a fair description of the known low-energy level densities of the nuclei which enter the calculations.

El

N U C L E A R REACTIONS 31p(p, %) and 31p(p, ~i), Ev = 8.4 to 11.8 MeV; measured ~r(E). 3zS compound states deduced average widths/~.

1. Introduction High resolution measurements

of excitation functions of nuclear reactions to

i s o l a t e d final states s h ~ w s t r o n g i r r e g u l a r f l u c t u a t i o n s w h e n t h e e x c i t a t i o n e n e r g y is sufficiently h i g h a n d i n d i c a t e t h a t a c o m p o u n d n u c l e u s in t h e r e g i o n o f s t r o n g l y o v e r l a p p i n g e n e r g y levels is f o r m e d . T h i s b e h a v i o u r was p r e d i c t e d by E r i c s o n 1,2) a n d others. By use o f t h e statistical t h e o r y o f n u c l e a r r e a c t i o n s , these a u t h o r s s h o w e d t h a t the a v e r a g e level w i d t h F c a n be d e t e r m i n e d f r o m a statistical analysis o f such * Based on work performed under the auspices of the U.S. Atomic Energy Commission. 465

466

H.K. VONACHet al.

excitation functions. Furthermore, it is possible to calculate 3) the level density of the compound nucleus from the experimentally determined level width F. As a contribution to the study of the mass and energy dependence of the level width F, excitation functions of the reaction 31p(p, ct)28Si were measured. Preliminary results of these measurements have already been published 4). In the meantime, two more investigations of cross-section fluctuations in the 31p(p, e)28Si reaction have been reported. Leachman e t al. 5) investigated this reaction at both higher and lower excitation energies than our measurements and Dallimore and Allardyce 6) covered the same energy range as our experiments. Dallimore and Allardyce 6) compared the fluctuation amplitudes and their dependence on angle and final state with the predictions of the fluctuation theory and found good agreement. Both groups of authors determined the level width F. However, only Leachman e t al. s) compared this quantity with statistical-model calculations. They used a special form of the nuclear level density proposed by Gilbert and Cameron 7), namely a constant temperature level density at low excitation energies and a Fermigas type level density at high excitation energies. The purpose of this paper is to give a more detailed report of our experimental results and to compare the level width F with statistical-model calculations based on the Fermi-gas model instead of the Gilbert-Cameron model.

2. Experimental procedure Excitation functions of the reaction 31p(p, g)28Si were measured with the Argonne T a n d e m Van de Graaff in proton energy intervals of 8.37 to 9.00 MeV, and 10.00 to 11.77 MeV in 10 keV steps. A phosphorus target of approximately 25 /~g/cm 2 thickness was prepared on 100/~g/cm 2 carbon backing with the Argonne National Laboratory mass separator t The entrance and exit slits of the beam analysing magnet, which has a radius o3 curvature of 86 cm were set at 0.9 m m in these experiments. The energy resolution of the proton beam was estimated from experiments with the A N L broad-range single-gap magnetic spectrograph. The width of the observed peaks depends on the energy width of the beam, the thickness of the target and the aberration of the spectrograph** which is 0.075 percent (5 keV at Ep = 7 MeV). Peak widths of 6 keV ( F W H M ) were observed for inelastic proton groups of 6 to 8 MeV from (p, p ' ) reactions with targets of 10/zg/cm a (1 keV) and bombarding energies of 9 to 11 MeV. Therefore, the resolution of the beam is assumed to be of the order of or better than 5 keV for bombarding energies of approximately 10 MeV. The alpha particles from the reaction were detected by surface-barrier solid-state detectors in a 45 cm scattering chamber. In order to discriminate against protons, the detectors were biased to just stop the ground-state alpha particles. Deuterons t We thank J. Lerner for preparing the alp target. t* The aberration of the spectrograph is determined by the size of the beam spot on the target. It is 0.075 percent with a beam spot of (0.5 x4.0 mm)2.

LEVEL WIDTH OF 32S

467

and tritons were energetically prohibited due to their large negative Q-values. Silicon N-type detectors were used with specific resistivity of 600 to 2000 o h m - c m and active area 50 m m 2. Collimators of 0.63 cm diameter were placed in front of the detectors, and the detectors were covered with thin nickel foils of 100/~g/cm 2 in order to exclude low-energy electrons. The detectors were placed in the reaction plane and subtended an angle of 7 degrees. 3. Results 3.1. E X C I T A T I O N

FUNCTIONS

Excitation functions for the g r o u n d (0 +) and first excited (2 +) levels in the residual nucleus 28Si were measured in 10 keV steps for two p r o t o n energy regions. Measurements were made in the region 8.37 to 9.00 MeV at laboratory angles 30 °, 60 °, 90 ° , 120 °, 150 °, 170 ° and 175 °, and in the region 10,00 to 11.77 MeV at 39 °, 69 :', 81 ° , 111 °, 141 °, 161 ° and 175 ° . [n a third experiment, measurements of the excitation functions were repeated in the energy region 10.00 to 10.50 MeV. The results f r o m the last two experiments at 141 ° are displayed in fig. 1 and illustrate the reproducibility of the experimental measurements. The differences in the two curves in fig. 1 are approximately within the limits of the errors due to counting statistics, the latter being 3 to 5 percent for the g r o u n d state and 1 to 2 percent for the first excited level. The excitation functions are displayed in figs. 2 to 4. They are in excellent agreement with the excitation functions reported in ref. 6). 3.2. S T A T I S T I C A L

ANALYSIS

Since the fluctuation amplitudes and cross correlations were analysed extensively in ref. 6) and f o u n d in satisfactory agreement with theory we shall not repeat our similar analysis in this paper. Values o f the above mentioned quantities determined f r o m our data are available in ref. 8). We restrict the following discussion to the determination of the level w i d t h / " f r o m our experimental excitation functions. A u t o correlation functions, defined by the relation =

a

a

were calculated for all excitation functions where a(ei) is the cross section at energy el, ( a ( e l ) ) the cross section averaged over an energy interval large c o m p a r e d to /" and centered at ei and the bracket ( ) means an average over excitation energy. In order to study the energy dependence o f F, the total energy interval covered in the experiment was divided into 3 parts, corresponding to b o m b a r d i n g energies of 8.37 to 9.00, I0.00 to 10.90 and 10.90 to 11.77 MeV. Separate auto-correlation functions were calculated for these 3 excitation-energy regions, resulting in a total of 42 auto-correlation functions.

468

H.K. VONACH et al.

1600 I

12001~

!

Q O

"-

(n uJ o9 .J

800

Q.

31p (p, ao)2a Si EXCITATION FUNCTION 1,41 DEGREES

V

4OO

0

I0.00

I 10.20

I

I

I

I

I

10.,:1-0 10.60 10.80 I 1.00 I 1.20 PROTON ENERGY, MeV

8000

I

I 1.40

I I 1.60

1.80

nt p(p, "l) 2eSi

EXCITATION FUNCTION 141 DEGREES

600C

0 0

4000

200C

0 [ IO.OO I0.20

I

10.40

I

I

I

I

t0.60 10.80 II O0 I I.;:'0 PROTON ENERGY, MeV

I

II .40

I

I 1.60

I 1.80

Fig. 1. Excitation f u n c t i o n s for the 81p(p, ~0)~sSi a n d 31p(p, m~)2sSi reactions f r o m two different experiments.

LEVEL WIDTH OF ass

469

In these relatively small excitation energy regions the change of the average cross section 6) with excitation energy may well be approximated by a linear dependence of (~(~i)) Oll the excitation energy ~ [see for example figs. 3 and 4 of ref. 6)]. 31p(p,a)ZaSi

~

. I s l EXC. STATE

G R O U N D STATE

i

5o

)o

i

Z FF OC I--

g

5 I

(#) 0 tY o

OL= 6 0 °

i

8.4

8.6

8.8

9.0

8.4

816

88

i

91o

PROTON ENERGY, MeV

Fig..2. Excitation functions for the alp(p, %)zaSi and alp(p, 0q)zaSi reactions in the p r o t o n energy range f r o m 8.37-9.00 MeV.

In the calculation of R(&), therefore, it was assumed that

+ b(ei- (~)),

(2)

where g, is the excitation energy of the compound nucleus, and (g) the excitation energy of the compound nucleus at the center of the excitation energy range used for calculation of the autocorrelation function. The slope b was derived from the experimental excitation functions in the following way. A straight line was fitted to each

I 0.4

10.8 I 1.2 PROTON ENERGY, MeV

o

I 0.0

I 1.6

Fig. 3. Excitation functions for the 81p(p, ~0)28Si reaction in the proton energy range 10.00 to 11.80 MeV.

o

¢.~

Z O

:i

F i g . 4. E x c i t a t i o n

10.0

m

I-.°

Z

I-.--

t--

g

)n~

=

OL = 141 °

= I 75 °

>-

z

-

Slp(p,Clo)~'ilSi

Slp(p,Clo)~'ilSi

o

10.8

=

i

0L

i

81

e n e r g y r a n g e 1 0 . 0 0 t o l 1.80 M e V .

f o r t h e 31p(p, e l ) 2 8 S i r e a c t i o n

=

i

0L = i1[o

@L 141°

I 1,2

PROTON ENERGY, MeV functions

10.4

31p(p,al)28Si

in the proton

I 1.6

i

r

i

>

0

<

=

LEVEL WIDTH OF 32S

471

i n d i v i d u a l excitation function b y the least squares m e t h o d . Then an average slope b was c a l c u l a t e d f r o m all excitation curves in the same excitation energy range. These values o f b were used for the c a l c u l a t i o n o f the a u t o - c o r r e l a t i o n functions, the a p p r o p r i a t e b value for each c o r r e s p o n d i n g excitation energy range. F o r an infinite averaging interval, t h e o r y 1) predicts the a u t o - c o r r e l a t i o n function to be L o r e n t z i a n with half-width F, 1 R(~5) = c o n s t F2 q_62 "

(3)

I f finite-range effects are neglected, the level width F is o b t a i n e d as the half-width o f the e x p e r i m e n t a l l y d e t e r m i n e d a u t o - c o r r e l a t i o n functions. TABLE 1

Half width of autocorrelation function calculated from excitation functions for the sip(p, s0) and zip(p, cq) reactions at several angles and bombarding energy ranges Ev (MeV) reaction zip(p, 0%)

a l p ( p , ~l)

8.37-9.00 /~ (keV)

angle (deg) 30 60 90 120 150 170 175 30 60 90 120 150 170 175

53 54 16 60 16 / ~

/ J

25 16 27 26 34 35 28 T'~=32

angle (deg)

10.00-10.90 -P-l(keV)

10.90-11.77 /'~ (keV)

39 69 81 111 141 161 175

39 44 57 28 28 39 38

31 59 57 36 25 21 23

39 69 81 111 141 161 175

58 47 36 26 27 31 71

46 33 32 84 28 35 33

T~=41

T'~=39

T h e values o f the h a l f - w i d t h F4 f r o m all o u r a u t o - c o r r e l a t i o n functions are listed in table 1. Since significant systematic differences in the half-widths for different angles a n d final states were neither observed n o r expected theoretically, average halfwidths F4 were c a l c u l a t e d for each excitation energy range f r o m all 14 a u t o - c o r r e l a t i o n functions. C o l u m n 1 o f table 2 gives the average half-width f ~ d e t e r m i n e d in this w a y as a function o f excitation energy. The errors i n / ~ were calculated in the usual w a y f r o m the m e a n square deviations of the i n d i v i d u a l F~ values f r o m -P~v. A s a l r e a d y m e n t i o n e d , the level-width F is simply equal to the half-width F~ o f the a u t o - c o r r e l a t i o n f u n c t i o n if the l a t t e r is calculated by averaging over an infinite energy interval. The necessary use o f finite energy intervals has the consequence o f c h a n g i n g this simple r e l a t i o n between F a n d the half-width F~ o f the a u t o - c o r r e l a t i o n

H.K. VONACI-Iet al.

472

curve and introduces r a n d o m deviations o f the observed F~ values from their expectation values. Both effects have been calculated by Gibbs 9) and B/Shning lo) as a function of the so-called effective n u m b e r n of independent cross sections,

AE

n = --, roe

(4)

where AE is the length of averaging interval 2). The values o f n for our experiments are given in table 2. Values of the level width £ were calculated from the appropriate average half-width /~{ and value o f n according to eqn. (52) of ref. 10). Values of the expected mean square deviation AF~theoret" o f the observed £~ from the average value due to the finite-range effects were also determined according to eqn. (52) of ref. to). The values o f these quantities as well as the experimentally observed mean square deviation AF{exp" are given also in table 2. The values of AF.~xp" are only slightly larger than A['_~th. . . . t . " Thus most of the observed rather large deviations of the individual half-widths f r o m / ~ are probably due to finite range effects. TABLE 2 Values of the level width/' of saS derived from 8~p(p, c~)excitation functions averaged over angle and final states of the residual nucleus Bombarding Excitation energy energy (MeV) (MeV)

f'~ (keY)

n

f' (keV)

ZIF.t-exp (keY)

~l_P+,h (keY)

8.37- 9.00 17.8 -18.40 10.00--10.90 19.4- 20.25 10.90-11.77 20.25-21.10

324-4.5 41-t-3.5 39i4.5

7 8 8

38=t=5 47~4 45J:5

18.4 13.0 16.7

12 14 13

n = number independent cross sections calculated from eqn. (4). A/'~ exp = ~/(F~--T~) 2 averaged over the 14 values of/~k in each excitation energy range. /[/~th theoretical expectation value of zl/~x if error is caused by finite range effects only. ~

4. Statistical theory calculation of decay widths

4.1. GENERAL REMARKS In this section, the values of F derived in the previous section will be c o m p a r e d to the predictions o f the statistical model of nuclear reactions. As discussed previously 3), the calculated values o f F depend on the level density of the c o m p o u n d nucleus and the level densities o f all residual nuclei in the excitation energy ranges reached by the various decay modes o f the c o m p o u n d nucleus. Obviously it is not possible to determine uniquely all of the level densities o f the residual nuclei and c o m p o u n d nucleus f r o m the measured width F at a particular c o m p o u n d excitation energy. However, a considerable a m o u n t o f information 1~) exists already on the level densities o f all the different nuclei involved in the calculation of the level width. F r o m high resolution magnetic spectrograph measurements of particles emitted in nuclear reactions and studies o f (p, ~) and (~, ?) resonances, the level densities of

LEVEL W I D T H OF aSS

473

the important residual nuclei 31p and 28Si and the compound nucleus 32S are known up to an excitation energy of about 10 MeV. In the following section, we shall use this information to determine the free parameters of the various proposed level-density formulas and then calculate the level width F with these various level densities as a function of energy. Comparisons of these F values with those from fluctuation analyses show which of the proposed level density laws, if any, are consistent with the energy-dependent level-width at high excitation energy as well as the low-energy level-density data. 4.2. RELATION BETWEEN LEVEL DENSITY AND DECAY WIDTH OF HIGHLY EXCITED NUCLEI The width F b of compound nuclear states of spin J and excitation energy Uc for decay by emission of particles b leading to a residual nucleus B, is given by 1) 1

( v . ..... dUB Z Tu,~(eb) F~(Uc) - 2nCoc(Uc, J) 30 ,2 12+/21

x

IS2+Ibl

Z

Z

o),(U,, I,),

(5)

S2=[J-121 IB=lSz--Ibl

where I b and I B are the spins of the particles b and B, respectively, $2 the exit channel spin, l 2 the orbital angular momentum of b, and Tb/2(gb) the transmission coefficient of particle b with orbital angular momentum 12 and kinetic energy ~;b. The quantities e~c(Uc, J ) and o)a(Ua, Ia) are the energy and spin dependent level densities of the compound nucleus C and the residual nucleus B, respectively. The width F determined from fluctuation analyses as described in sect. 3.2. is a weighted average of the decay width for different spin states F s over the various compound spin states. These weighting factors are essentially the contributions of the various compound spin states to the cross section of the final state under consideration. These weighting factors, and therefore F itself, are in principle dependent on excitation energy, angle 0 and the spin and parity of the final state and have been evaluated by Gadioli et al. 12). Most experimental values of F do not show significant differences for different angles and final state spins 13) and we conclude that the dependence of the weighting factors on these variables is very weak. Therefore, we have not used the very complicated weighting factors of ref. 12) in order to calculate F values for comparison with experimental data. Instead it is assumed that the measured width is equal to the average width of the compound nucleus defined by the simple relation

F

~ as s

where a s is the cross section for the formation of the compound nucleus with spin

H.K. VONACHet al.

474

J at an excitation energy Uc. This cross section is given by

trj(Uc ' j)

=

7r; z(2J + 1)

+/AI

IJ+Sll yall( l) '

( 2 1 . + 1 ) ( 2 I A + 1 ) S =l'a-'AI 'l=lJ-Sll

(7)

where ;~1 is the de Broglie wavelength of the entrance channel, I A the spin of target nucleus, I a is the spin of projectile, $1 the spin of entrance channel given by $1 = IA+Ia, Ta~l(ea) is the optical model transmission coefficient of particle a with orbital angular momentum l~ and channel energy et and e 1 is the c.m. energy of the entrance channel. 4.3. CHOICE OF LEVEL DENSITIES AND LEVEL DENSITY PARAMETERS FOR THE CALCULATIONS OF THE LEVEL DENSITY As shown in fig. 5, the value of F b, and thus F, depends on the level densities of the compound nucleus 32S and the residual nuclei 31p, 31S, 2sSi and 30p reached by proton, neutron, s-particle and deuteron emission, respectively. The Q-values for the emission of the various particles are such that only proton and s-particle emission give important contributions to the total width (see fig. 7). Thus, we restrict ourselves to the discussion of the level densities of the three nuclei 32S, 31p and 2ssi. The available experimental information on the level density 11) of these nuclei is shown in fig. 5. The low energy level densities were taken directly from ref. 11) and the level densities based on the various resonance data were calculated as described in the appendix. The error bars on the resonance data include both the uncertainty in the spin cutoff factor o- (50 to 100 ~ rigid body moment) and the finite number of levels (assuming a Aco/co = 1/x/N due to the finite number of N observed levels). A number of forms for the energy dependence of the level density have been proposed which can, in principle, be used to calculate the width F as a function of excitation energy. However, the parameters must be chosen to fit the experimental level density data of fig. 5. Several different forms of the level density are discussed in the following sections. 4.3.1. The constant temperature model. This empirical model assumes an exponential increase of the level density with excitation energy and gives a good description of the level density of light and medium weight nuclei at low excitation energies (0-10 MeV). However, it predicts the level width to be almost independent of excitation energy. The results of Leachman et al. 5) show that the constant temperature model is not valid for higher excitation energies and therefore no calculations of F were made with this model. 4.3.2. The shifted Fermi-gas model. In this widely used model, the level density is assumed to have the form co(U, J) =

a~(2J+l) 24x/~e~(U+t_A) 2 exp ( 2 a ½ ( U - A ) ~ [ - J ( J + l ) / 2 c t ] } ,

(8)

----------__

Fig. 5. Experimental

data

OR the level densities

/+stograms

Fermi-Gas-Model

-Model

:&p

o=A/lO

o-A/i

o--A/9.

~MQ

/QVQt denslt~es

2A = v

-

S

32

model,

by countmg

Fermi-gas

dQtQrm/ned

A3’P =0, AzBS, z +3Mo( A3*S =+2MeV , 2A - 4MeV

of a1P, % and %i and predictions of the normal shifted gas model and the Gilbert-Cameron model.

Convenlion&ShIfted

@

Fermi-Gas

FQrml-Gas-Model

Back-shIfted Convent/ona/-shjfled

@

G//berl-Cameron-Model

@

@

---,

the back-shifted

/ndw/duot

Fermi-

--1----

2 VI

476

H.K. VONACHet al.

where, U-A

= at2-t,

(9)

and a is the Fermi-gas constant, t the thermodynamic temperature, and e = I/h 2, where I is the nuclear moment of inertia. The energy shift A used to describe the difference in level density between odd, odd-mass and even nuclei is assumed to be zero for odd nuclei and positive for odd-mass and even nuclei. In most cases, this model does not give good fits to both the low-energy level count data and the resonance data for any value of the level density parameter a. This is illustrated in fig. 5 for the present data. Nevertheless, some calculations were carried out with this model in order to demonstrate the necessity of improved models. 4.3.3. The back-shifted Fermi-gas model. A better description of the level density at low excitation energy is obtained if both a and A are treated as free parameters and determined by a fit to the experimental data. Since this leads to a shift of the fictive Fermi-gas ground state opposite in direction to the shift introduced for description of the odd-even effect in the shifted Fermi-gas model, it is called the backshifted Fermi-gas model 14). The level density predicted by this model is shown in fig. 5 for a special set of parameters, a = A/9 and a back-shift of about A. A reasonable fit to the experimental data is obtained for all three nuclei. 4.3.4. The Gilbert-Cameron model. This model assumes the level density to be given by the constant temperature model at low excitation energies and by the shifted Fermi-gas model at high excitation energies. As shown in fig. 5, it gives about as good a fit to the experimental low-energy level density data as the back-shifted Fermi-gas model. It has already been used by Leachman et al. 5) to calculate F and thus, we have not made any new calculations. 4.4. CALCULATION OF THE LEVEL WIDTH With the level densities discussed in the previous section, numerical values of the width F s for J up to 17 and of the average width F as a function of excitation energy were calculated by means of a F O R T R A N Program based on eqns. (5)-(9) with the SDS computer of the University of Washington. In this program the sums in eqn. (5) were evaluated up to l 2 = 17 and $2 = 14; the integration over the residual excitation energy Un was performed by summation over 0.5 MeV intervals using Simpson's rule. The optical model transmission coefficients used in the calculations were computed with the program ABACUS-2 15) revised for the A N L CDC-3600 computer. The optical model parameters were from Bjorklund and Fernbach for neutrons 16), Perey for protons 17), Huizenga and Igo for alpha particles 18) and Melkanoff et al. for deuterons 19). 4.5. RESULTS AND DISCUSSION In fig. 6 the results of the calculations of F are compared with the results of our measurements and those of ref. 5). Calculations were performed with both the

477

LEVEL W I D T H OF 82S

shifted and the back-shifted Fermi-gas model level densities with parameters a and A corresponding to the various level density curves displayed in fig. 5. In addition, the E values calculated in ref. s) with the Gilbert-Cameron model are also given. The partial widths F~, £] and £~ are shown in fig. 7 as a function of J for the backshifted Fermi-gas for one particular excitation energy. The spin distribution of the compound nuclei at the same excitation energy is shown in the lower part of fig. 7. F r o m this figure two important conclusions are drawn: (i) The J-dependence of £ in the range J = 1 to 4, which contains more than 85 ~o of the compound spin

1 i~°° [

1@

j~@

cot experiment •

-1 .I

j"

.I / /

I"

k

/ /

/

/

/

lO

,/

/

/"

/ 14

/

Gilbert-Cemeron-Model (as calculated in ref 5)

/ @ / ®

/

/

~ ref.5 (overage of Fp iT.o and Fpo~I)

/

/

I

/

, /

///

/

© i 16

Beck-shifted Fermi -Gas -Mode[ e=.g/9,A31P=O, 4ZSsi =+3MeV,A32S =+2MeV Conventionol shifted Fermi- Gas-Model o=A/8 24 =4 Mek Conventionel shifted Fermi-Gas-Model I'#

2o

J2

2".

2~ U(MeV)

o--A/tO 24=4h~V 2'e --

3o --

~'?

]~

Fig. 6. Energy dependence of the decay w i d t h / ' for 8~S. The calculations were made with the Fermigas form of level density using a rigid-body m o m e n t of inertia.

distribution is rather weak. This confirms the assumption made in sect. 4.2. that the exact averaging procedure is not critical for the calculation of F, ve; (ii) F r o m fig. 7 it is clear that r ~ gives by far the most important contribution to r and thus F is essentially determined by the level densities of the residual nucleus 31p and the compound nucleus szs. These two conclusions are valid also for other excitation energies and level densities for which calculations were performed.

478

u . K . VONACH et al.

It is now possible to judge the quality of the various level-density models by simultaneous inspection of the degree of agreement between the values ofFex p and Fc,~c and between the theoretical and experimental values of the level density at low excitation energy for 31p and 3zS. F r o m these comparisons we obtain the following results for the various models. 100

(e)

T

pjj totol

Qa

r~

G 70

U

T

0 "014

(b) 03

I

0,2

J 0

2

4

6

8

Fig. 7. (a) Spin dependence is s h o w n o f the width _f'jTotal a n d the partial w i d t h s / ~ / p , I ) n and F j ~ o f 32S at 20.9 M e V excitation energy. These curves were calculated with the back-shifted Fermi-gas f o r m o f level density a n d a rigid-body m o m e n t o f inertia. (b) Spin distribution is plotted o f the c o m p o u n d nuclei o f 82S at 20.9 M e V excitation energy f o r m e d by p r o t o n absorption on 8ip.

4.5.1. The conventional shifted Fermi-Gas model. As shown in figs. 5 and 6, no values of the level density parameter a can be found which gives a good fit to both the experimental level densities at low excitation energy and the experimental F values.

479

LEVEL W I D T H OF 32S

I f a = A/8 is chosen, we get a r e a s o n a b l e a g r e e m e n t with level densities a r o u n d 10 M e V derived f r o m r e s o n a n c e data, however, the c a l c u l a t e d widths F are much smaller t h a n the o b s e r v e d ones. I f the p a r a m e t e r a is r e d u c e d to A/IO we get a p p r o x i m a t e a g r e e m e n t with our o b s e r v e d w i d t h F. H o w e v e r , in this case the p r e d i c t e d level density does n o t agree at all with e x p e r i m e n t a l low energy data. 4.5.2. The Gilbert-Cameron model. A s a l r e a d y p o i n t e d out in ref. 5), this m o d e l with the p a r a m e t e r s chosen to fit the low-energy a n d resonance level-density d a t a , also predicts values o f F which are in r e a s o n a b l e a g r e e m e n t with the e x p e r i m e n t except at the highest energies. Thus, this m o d e l seems to give an a d e q u a t e d e s c r i p t i o n of the nuclear level density up to at least 20 MeV for the nucleus 31S. 4.5.3. The back-shifted Fermi-gas model. As shown in figs. 5 a n d 6, fair a g r e e m e n t (of a b o u t this same quality as for the G i l b e r t - C a m e r o n m o d e l ) is o b t a i n e d for this m o d e l for b o t h the e x p e r i m e n t a l F values and level densities at low excitation energies. A g a i n the a g r e e m e n t with e x p e r i m e n t a l F values is p o o r for the high energy p o i n t o f ref. 5). Thus, we can s u m m a r i z e the results o f the calculations o f F in the following way. The calculated widths F o f 32S a r o u n d 20 MeV are in g o o d a g r e e m e n t with experim e n t a l results for b o t h the G i l b e r t - C a m e r o n and back-shifted F e r m i - g a s model. These two m o d e l s give s i m u l t a n e o u s l y an a d e q u a t e d e s c r i p t i o n o f the level density in the 0-10 M e V range, whereas, such a g r e e m e n t c a n n o t be o b t a i n e d with the conventional shifted F e r m i - g a s model. However, all m o d e l s p r e d i c t a larger increase o f F in the 20-30 M e V range t h a n o b s e r v e d experimentally, a fact which c a n n o t be exp l a i n e d at present. Appendix

I n this section we describe the c a l c u l a t i o n o f the level densities o f 31p, 32 s a n d 2sSi at a b o u t 9-12 M e V excitation energy f r o m the average spacing of (p, 7), (e, 7) a n d TABLE 3

Charged particle resonance data for compound nuclei 2sSi, 31p and 32S and total level densities derived from this data Compound nucleus

Reaction

Excitation energy (MeV)

2ssi

2rAI(p, 7)

31p

24Mg(c~,y) and 27Al(p, ~0) 8°Si(p, ~)

32s

31p(p, y) 31p(p, s0)

Number of resonances

Spin and parities of resonances

12.45 to 12.95 12.0 to 12.5

21

0 =, 1±, 2±, 3±, 4 ±, 5~, 60+, 1-, 2+, 3-, 4 +, 5-

8.4 to 9.2 10 to 11 10 to 11

34 26 14

10

O'rigid 60rigid (levels/ MeV)

0"½rigid (0½ rigid (levels/ MeV)

2.8

45

2.0

43

2.8

44

2.0

40

½±, :~±,~±, ~}-

2.85

69

2.0

50

0 ±, 1±, 2±, 3~, 40+, 1-, 2+, 3-

2.9 2.9

37 45

2.05 2.05

28 31

Assuming all resonances with l = 0, 1, 2, 3 and no l > 4 resonances are observed.

H.K. VONACHet al.

480

(p, e) resonance data which is compiled in ref. 11). C o l u m n s 1 to 5 of table 3 summarize this i n f o r m a t i o n for the nuclei in question. In the case of the (p, 7) reaction, excitation energy ranges were selected c o r r e s p o n d i n g to a p r o t o n energy of 1 to 2 MeV. As discussed in ref. 7), one can then assume that all resonances with l = 0, 1, 2, 3 are observed and no resonances with higher l values are observed because of the rapid decrease of the p r o t o n t r a n s m i s s i o n coefficients with a n g u l a r m o m e n t u m l. F r o m the density of resonances O)~es the total level density ~o was calculated from the relation co~es :

co 2 ! 2 J + 1) exp J,~ 40-2

I -J(J + 1)7 20-2

(10)

] '

by s u m m i n g over the spins a n d parities allowed for the resonances. There is some uncertainty a b o u t the values of a to be used in eqn. (10). However, it can be assumed that the actual 0- values correspond to a nuclear m o m e n t of inertia of 50 to 100 percent of the rigid-body value. Therefore, the values of 0- were calculated from the Fermi-gas model using the parameters a a n d A listed in fig. 5 for m o m e n t s of inertia of 50 a n d 100 percent of the rigid-body values. The total level density was calculated for both sets of spin cut-off factors a n d the results are given in table 3 a n d fig. 5.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

T. Ericson, Ann. of Phys. 23 (1963) 390 D. M. Brink and R. O. Stephen, Phys. Lett. 5 (1963) 77 H. Vonach and J. R. Huizenga, Phys. Rev. 138 (1965) B1372 A. A. Katsanos, H. K. Vonach and J. R. Huizenga, Bull. Am. Phys. Soc. 9 (1964) 667 R. B. Leachman, P. Fessenden and W. R. Gibbs, private communication P. J. Dallimore and B. W. Allardyce, Nucl. Phys. A108 (1968) 150 A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 A. A. Katsanos, Argonne National Laboratory Report 7289 (1967), unpublished W. R. Gibbs, Phys. Rev. 139 (1967) B1185 M. B0hning, Max-Planck Institut ffir Kernphysik Heidelberg Jahresbericht (1965), p. 105 P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1 E. Gadioli, I. Iori, A. Marini and M. Sansoni, Nuovo Cim. 44B (1966) 338 See, for example, ref. 8) or W. Von Witsch, P. Von Brentano, T. Mayer-Kuckuk and A. Richter, Nucl. Phys. 80 (1966) 394 J. R. Huizenga, H. K. Vonach, A. A. Katsanos, A. J. Gorski and C. J. Stephen, unpublishedresults E. H. Auerbach, ABACUS-2 (Revised Version), Brookhaven National Laboratory, Upton L. I., New York (Nov. 1962), unpublished F. Bjorklund and S. Fernbach, Phys. Rev. 109 (1958) 1295 F. G. Perey, Phys. Rev. 131 (1963) 745 J. R. Huizenga and G. Igo, Nucl. Phys. 29 (1962) 462 M. A. Melkanoff, T. Sawado and N. Cindro, Phys. Lett. 2 (1962) 98