Excitation of the even-even nuclei 24Mg, 28Si and 32S by neutron scattering at incident energies between 7 and 14 MeV

Nuclear Physics @ North-Holland

A440 (1985) 35-61 Publishing Company

EXCITATION BY NEUTRON

OF THE EVEN-EVEN NUCLEI 24Mg, 2xSi AND 32S SCATTERING AT INCIDENT ENERGIES BETWEEN 7 AND

M.

ADEL-FAWZY*,

H. FGRTSCH,

14MeV

D. SCHMIDT,

D. SEELIGER

and

T. STREIL

Technische Uniz-ersifiitDresden, Se&ion Physik, 8027 Dresden, German Democratic Republic Received 25 July 1983 (Revised 1 I January 1985) Abstract:

Neutron scattering on 24Mg, “Si and “S was studied in the incident energy range from 7 to 14 MeV. In the experiment natural samples and the time-of-flight technique were used. The theoretical analysis was extended including other experiments to bombarding energies around 14 MeV. The excitation of the first 2+ levels shows for these nuclei, despite their quite different nulcear structure, some similarities. For 24Mg also higher-lying states up to E, = 6.01 MeV have been investigated. The sum of compound-reaction and collective-model contributions reasonably reproduces all the experimental data.

E

I

measured REACTIONS ‘;Mg; ‘?Si, “S(n, n’), E =7-14MeV; deduced reaction mechanism. Coupled-channel calculations.

NUCLEAR

trfE

, E n,s @); j

1. Introduction Low-lying nuclear states are known to be strongly excited in nucleon-induced reactions due to their collective nature. Such collective excitation can be understood simply

as a result

of stationary

or dynamic

deformation,

i.e. a deviation

from the

spherical shape. The deformation parameters can be derived from an analysis of the differential cross sections, but in that case their absolute values also depend on other model parameters, such as the potentiai parameters of the averaged nulcear field. At the present time, neutron scattering can be used for such investigation in competition with charged particles because of the progress in experimental techniques as well as the availability of neutron scattering data, especially for separated isotopes. However, the deformation parameters extracted from scattering experiments with charged particles and neutrons are generally different because of the different interacting averaged field for both cases. This was the reason for a systematic study of the behaviour of the &,,/&,, ratio for the closed-shell even Sn isotopes (Z = 50) [ref. ‘)] and N = 50 isotones 88Sr, 90Zr and 92Mo, published in ref. ‘). In the last few years, many neutron scattering experiments in the MeV region were concentrated on the investigation of nuclear deformation effects. This requires a small influence * Permanent

of reaction

address:

AIexandria

components University,

via the compound Alexandria 35

(Egypt).

nucleus,

i.e. a relatively

36

M. Adel-Fawzv et al. / 24Mg, z8Si, “S(n,

n’)

high incident neutron energy with respect to the mass-number important question is the choice of the optical potential parameters.

region 3-5). An As outlined in

ref. ?, there are several possibilities for extracting generalized optical potential parameters to characterize the averaged nuclear field. As shown for even MO ‘) and Se isotopes *) strong coupling of the first 2+ levels to the ground states has to be included into potential parameters analysis. Another general problem of such investigations is the extraction of the quadrupole deformation parameters which can be chosen by comparison with charged-particle experiments such as (p, p’), ((~,a’), (d, d’), (e, e’), (3He, ‘He’) or from transition probabilities B(EA, O++ A “). But as outlined above, this method is questionable for nuclei near closed shells. The problem is discussed more in detail in ref. ‘). Furthermore, the magnitude of the quadrupole deformation p2 depends on whether a hexadecapole component p4 is taken into consideration, or not “). Such investigations at several bombarding energies were substantially extended into the rare-earth and actinide mass-number region 67’o), but in most cases only the first excited state was taken into account in the analysis. The excitation mechansim for the low-lying excited states in even-even nuclei is described in the present work by an incoherent sum of two components: the emission from the compound nucleus treated within the Hauser-Feshbach-Moldauer theory, and the direct excitation calculated within the collective model using the coupledchannel method. The models used influence each other because in both models the reaction channels which are not treated explicitly are considered as an absorption potential term. In order to take this effect into account, one can calculate the transmission coefficients for the compound-reaction part directly from the coupledchannel scattering matrix ‘I), or the absorption term in the optical potential has to be modified. This latter method has been used for the analysis in the present work. It is well known that at bombarding energies near 14 MeV, for the nuclei studied here, the direct excitation of low-lying states dominates 12-14) and the compoundreaction part plays a minor role. On the other hand, at incident energies near 3 MeV a direct-reaction part is still of the order of lo-20% [refs. ‘*“)I of the partial cross sections for these states, though the absolute value of compound-reaction component is much higher than at 14 MeV. Thus, the question concerning the competition of both components in the energy range between 3 and 14 MeV and its quantitative dependence on incident energy arises and will be investigated in this work. Furthermore, for 24Mg, “Si and 32S the amount of direct-reaction contribution is intluenced by the coupling to higher-excited states explicitly representing reaction channels. This fact stimulated our aim to measure and theoretically analyse as many high-lying states as possible, in the cases of 24Mg and 28Si up to an excitation energies of 6 MeV. The light even-even nulcei 24Mg, 28Si and 32S have been selected because they are known to have quite different structure, i.e. excitation mode. Our analysis was mainly concerned with looking for changes of the reaction mechanism with fast neutrons in a wide energy range.

M. Adel-Fawzy et al. / 24Mg, 28Si, 32S(n, n’)

37

2. Experimental procedure and results Differential cross-section measurements were performed using the eight-detector neutron time-of-flight facility at the 10 MeV Rossendorf tandem accelerator. The experimental arrangement is described extensively elsewhere 16) and therefore only a brief description will be given here. Fig. 1 shows an overview of the experimental set-up. Incident neutrons were produced using the ‘H(d, n)3He reaction. The deuteron beam from the accelerator was pulsed and bunched into bursts of repetition rate 5 MHz with a burst width of - 1.5 ns FWHM. The averaged pulsed current was typically 1 PA. The beam entered the 4 cm long deuterium gas cell through a 4.1 (*rn

remote

mem.

computer

Lk 32k f reader

Fig. I. Experimental

arrangement

12blt t

1 Ipuncher]

2Lbit

[

# djsplay]

at the 10 MeV Rossendorf tandem neutron spectroscopy 16).

used for the time-of-flight

fast

M. Adel-Fawzy et al. / 24Mg, 28Si, “S(n,

38

thick molybdenum and position,

foil “). The gas pressure

respectively,

were checked

was about

n’)

1.3 X 10’ Pa. The beam focus

by a four-segment

slit in front of the target

assembly. Neutrons

with energy ranging

from 6.8 to 12.0 MeV were scattered

from cylindrical

samples located at 0” with respect to the deuteron beam axis and 18.0 cm from the center of the gas target. The scattered neutrons were detected by an array of eight recoil proton detectors placed at 20” intervals. Each detector consisted of a 12.7 cm diameter, 3.8 cm thick NE-213 liquid scintillator, optically coupled to a FEU-63 photomultiplier tube. Each detector was housed in a heavy shield of paraffin and lead behind a 2 m long collimator of iron, oil and carbon. Eight 20 cm long iron shadow bars reduced the background caused by direct neutrons coming from the source. The flight path from the sample to each detector was -3 m. Data were collected using standard time-of-flight techniques with n/y discrimination to reject y-ray-induced events in the scintillator. The pulse-height bias for different measurements was set electronically corresponding to a neutron energy in the energy range between 0.8 and 1 MeV. The efficiency curves were taken from Monte Carlo calculations and checked with the well-known “*Cf method 18*19). An auxiliary stilbene scintillation detector with n/y discrimination was used, in the time-of-flight mode, for monitoring the primary neutron flux. This monitor

cn

&

2 0 1000

(a>

800

600 LOO 200 0 I

I

I

I

50

100

150

200

CHANNEL

Fig. 2. Typical TOF spectra taken from the 24Mg(n , n’) reaction at different angles and energies (the arrows indicate the expected energetic positions of the corresponding neutron lines; the hump around channel 115 in (b) is caused mainly by scattered deuteron break-up neutrons from the source).

M. Adel-Fawzy et al. / 24Mg, **Si, 32S(n, n’)

detector

was calibrated

for each measuring

run relative

39

to the main detectors

using

the same laboratory angle 20” with respect to the ‘H(d, n) neutron source. In this way the absolute normalization of the angular distributions was obtained. The samples

consisted

of natural

isotopic

compounds.

The shape of samples

was a full

or hollow cylinder. Dimensions and masses are presented in table 1. Measurements were completed over the angular range from 20” to 160” in 20” steps, and for 28Si additional runs at eight angles 15”, 30”, 50” to 150”. Yields for partial cross sections were obtained for isolated as well as overlapping peaks in the spectra by fitting of non-symmetric gaussian curves to the peaks including background. Fig. 2 gives an example for a TOF spectrum with results of fit procedures. Beside the monoenergetic ‘H(d, n) neutrons, also background neutrons from the source were scattered on the sample nuclei “) resulting in neutron lines and/or continuum within the measured spectra. These background neutrons were also taken

I

I

v,

I

I

I

I

I

I

Mg (n,n’l

2

E,,=11MeV

q=

2 u

LO0

LL= 3.02 m

3500

(b) t

3000

”

1 l.

2500

,

l .

l

.

2000

1500

1000

500

60

70

60

90

100

110

120

130

140

150

160

170

CHANNEL Fig.2 (cont.)

180

*

40

M. Adel-Fawzy et al. / 24Mg, 28Si, “S( n, n’) TABLE

I

Dimension and mass of the samples used Main isotope

%

-Mg “Si 32s

78.99 92.23 95.02

Inner diam.

Outer diam.

Mass

Ecml

[cm1

lcml

[al

3.0 3.0 3.0

0.0 0.0 1.0

3.0 3.0 3.0

39.36 43.53 36.30

Height

into account by the fit procedure, whereby the lines were separated from the scattering effect and the continuum was taken into consideration as non-linear background. Further aspects of this problem are outlined in detail in ref. 19). The net yields were corrected for the eftects of sample size, flux attenuation and multiple scattering. The corrections were introduced using the semi-analytical method proposed in ref. 20). This method requires the cross sections at incident energy and only averaged cross sections at the energy of the outgoing neutrons, i.e. resonance effects have not been considered. The corrections were always less than 20% of the measured cross sections resulting from the small size of all samples. The contribution of lines within the spectra resulting from scattered neutrons on 25.26Mgis assumed to be small. The number of these neutron lines versus the experimentally resolved energy interval is between 2 and 5 in the energy region investigated. Therefore, these neutron lines should give a certain “background” (see fig. 2) which is taken into consideration by the fit procedure. The first strongly excited 2” level in ?Ag is separated from the 2: state in 24Mg by one-half of the experimental FWHM. The fit to this 2: peak is done together with the elastic peak using the same peak parameters. Therefore, an additional unce~ainty following from interference of neutrons scattered on 25,‘6Mgis estimated to be less than 5%. Unce~ainties in the cross sections arise mainly from the following sources: counting statistics and background subtraction 510% (also greater for weakly excited states), energy dependence of detector efficiencies up to 7%, and normalization and geometry uncertainties 4%. The integral cross sections were obtained as usual by the Legendre polynomial fit procedure. The maximum order of polynomial expansion depends on the number of experimental points available. Numerical values of the pa~ial-differential cross sections as well as their Legendre polynomial expansion coefficients are published elsewhere 2’m23).In the present paper cross sections are analysed corresponding to the following levels: the 2: level in “Si and “S and the ground state, the 2:, 3:, 4: levels and the 4:/2: level group in 24Mg (see also fig. 3). The ground state and higher-lying levels for ‘*Si are described in another publication 24). For all three nuclides available 14 MeV data from the iiterature are also included in the interpretation.

M.

Add-Fawzvet ai. / 24Mg “Si 3

9

=s( n, n’)

41

8.113 ---06'

6.432 6.010 5236

-3'

5.413 ----

3+

3.778 ----

0'

2.230 -

2'

4.239 L.123 =$

1.779 -2' 1369

-

2+

32

28Si

S

Fig. 3. Level schemes of investigated nuclei; the full lines denote those levels which were measured experimentally in the present work (the 0+/6.69 state in “Si is weakly excited and was not included).

3. Excitation

of the first 2+ states in 24Mg, **Si and 32S

The model used in this analysis is the usual incoherent superposition of a compound-reaction contribution and a direct-interaction part. The range of bombarding energy above 7 MeV seems to be high enough, so that the number of open reaction channels shoutd diminish the influence of channel correlation. The compound~nucleus reaction part (HF) was calculated in the frame of the well-known Hauser-Feshbach theory including Moldauer’s width-fluctuation correction. The transmission coefficients were taken from the spherical optical model (SOM) where the optical potential has the following form: U(r) = -V&r,

Ro, ~,)+4iu,U:~~(r,

R,, a,)+

V~.&t_s~

&-,

R,, ao)

with f(r, Ri, ai)={1+cxpI(r-RiA”3)/ai]}-‘. Numerical calculations of the HF part were carried out using two computer codes. The code STAPRE 25) can calculate only integrated cross sections, but allows one to take into account a level continuum for higher-lying unknown levels in the different residual nuclei based on the backshifted Fermi-gas model. The code ELIESE ‘“) enables one to calculate angular distributions following from the HF model including a maximum of 30 known levels in the open reaction channels. In the case of 24Mg, only individual levels could be taken into account at bombarding

42

M. Adel-Fawzv. et al. / 24Mg, 28Si, 32S(n, n’)

energies up to 9 MeV. At higher incident energies, a level continuum had to be included. Thus, ELIESE results were normalized with the STAPRE calculations for incident energies from 7 to 9 MeV and used in the present work. In this energy range the ELIESE and STAPRE results are almost identical in the case of a constant absorption term W, = 7.76 MeV. Using the energy-dependent absorption term W, = OSE, calculations from both codes differ within 10% (g.s., 2:, 2;+4:) and 15% (3:, 4:), respectively. These deviations in the angle-integrated cross sections may be due to the different methods of numerical calculation of the transmission coefficients used [see refs. 25,26)].The uncertainty of the HF calculations is assumed to be in the same order of magnitude. This assumption was supported by recent systematic investigations with the HF model of all reaction channels in the energy range up to 20 MeV in connection with the new data evaluation for 28Si carried out by our group. Optical-model parameters (OMP) were not extracted by a fit to the elastic scattering data because of the comparatively small number of experimental points in the angular distributions (24Mg, 32S). The OMP used in this work are taken from different authors who have analysed elastic and also inelastic neutron scattering data together. The differences in the OMP used may reflect the different nuclear structures of the neighbouring light even-even nuclei considered. The OMP and other parameters used are arranged in table 2. The OMP for the (n, p) channels were taken from refs. 30*3’),for the (n, (u) channels from refs. 30.32),and the level-density parameters from refs. 33,34),respectively. As shown in fig. 4 in the case of 24Mg, the use of the OMP according to table 2 gives a quite reasonable description of the elastic scattering in the full energy range. TABLE 2

OMP and other parameters used in the calculations

OMP for the (n, n) channet V0 WeV1 R, lfml a, lfml W, W%l K [fm3

a,Ifml V, 0. EMeV

extracted at incident energy [MeVl ref.

49.68 1.17 0.60 7.76 1.09 0.69 7.t2

52.00

1.15 0.78 12.10

1.25

47.00 I .24 0.70 8.50

1.24 0.60 3.80 9 29)

112’)

0.47 4.90 10Z8)

+0.55 -0.05

+0.48 -0.30

+0.30 -

0.6E

8.5

deformation parameters P? P.l energy dependence of W,(E) [Mew 0.5E s 5.43

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

1000 7 MeV

1O~V

8Met/

-&CeHF ---SOM+HF --- HF

10

80

120

160

40

80

120

160

Fig. 4. Angular distributions from elastic scattering on z4Mg {see also text).

I

I

I

M. Adel-Fawzy et al. / “‘Mg, 28Si, “S(n,

44

n’)

The OMP were taken from authors who had analysed neutron scattering experiments at one point in the bombarding energy range investigated in the present work. Therefore,

only energy-independent

of this dependence

potential

depths

from our results was questionable

were reported.

An extraction

due to the limited

number

of

data points. But, a comparison of calculations with different potential depths V, ([from refs. 35,27)], d V/ V = 4% ) does not give any essential deviation in the angleintegrated elastic cross sections, but only some change in the shape of elastic angular distributions. Therefore, in our calculations we kept constant the OMP except W, which was assumed to be energy-dependent. The direct-reaction contribution (CC) is derived from the collective model resulting from introduction of a generalized deformed optical model due to

for vibrational

nuclei,

and

for rotational nuclei, with notation as defined in ref. ‘I). This leads to a set of coupled equations representing all open channels where the coupling is known to take into account multi-step contributions. The CC part was calculated with the code CHUCK 36) which is restricted to 8 reaction channels, 16 couplings and 189 coupled equations, respectively, but cannot take into account two-phonon excitation as well as asymmetric rotation modes. The deformation parameters (see table 2) were chosen after comparison of the values available from neutron as well as charged-particle experiments. By variation of the p2 values and comparison in magnitude with the 2: angular distributions the quadrupole deformation parameters pz could be fixed within f 10% concerning the values in table 2. The first 2+ states of the light even-even nuclei discussed here are known to show the greatest excitation strengths of all excited states. This can be obtained, for instance, from the magnitude of the angle-integrated cross sections at Eo= 12 MeV by subtracting some HF contributions. The remaining value for the 2: state is 5 or 10 times greater than the cross section to high-lying states (for 24Mg see figs. 10 and 13; for 32S and 28Si it is similar). In 24Mg, the 2: state is the first member of an almost “pure” rotational band. One indication is the ratio of excitation energies which follows the E,= Z(Z+ 1) rule. This structure has been used in analyses of other scattering experiments, such as (n, n’) [ref. 14)], (p, p’) [refs. 37*38)],(3He, ‘He’) [ref. 39)], (CX,a’) [ref. 40)], etc. In 32S, the first 2+ state can be understood as a one-phonon vibrational state, seen in the energetic spacing to the ground state and the two-phonon triplet O+-2+-4+. Using scattering of polarized protons at incident energy E,= 24.5 MeV [ref. 4’)], different deformation modes were investigated, such as vibrator, rotator, asymmetric

M. Adel-Fawzy et al. / 24Mg, 28Si, “S(n,

45

n’)

2LM~ (n,n,) Cl=- 1.3666 MeV

I’=

2+

-CC + HF . . . . . .... cc ---

C--N

.\

lI

I LO

I

I 80

I

I

120

I

I

160

I

I

LO

I

NW/' I

60

/-

I I

HF

I

I

120

I

160

O,,[degl Fig. 5. Angular

distributions

from inelastic

scattering on “‘Mg; from ref. 14).

W,(E) = 5.43 MeV = const;data n taken

rotator, and mixing of one- and two-phonon components. Nevertheless, in several vibrational excita(n, n’) [refs. “*‘“)I and (p, p’) [refs. 42*43)]studies the one-phonon tion is used for the description of the 2: state in 32S, and also in the frame of the analysis represented here. 28Si shows an intermediate behaviour: the energetic ratio E(4+)/E(2+) =2.6 is substantially smaller than that following from the rigid-rotator model (= 3.3); also a high hexadecapole component could be obtained. The question concerning the more different structure in 28Si is discussed in detail elsewhere ‘“).

M. Adel-Fawzy et al. / 24Mg, 28Si, 32S(n,

46

l

n’)

8.OMeV

,,,,,I,

--* A

L_,/

I _

20 40 60 80 100 120140160180

/C---r

,’

1

“5 ( n,nl ) 28Sillc Q = - 1.779 3” = 2 +

f

l

9.0MeV

--

**

l-1 ” 1 ” ’ i--i 0 20 LO 60 80 1001201401600

l12.0MeV -

” ” ” I-20 40 60 80 1001M140160180

OJdegl Fig. 6. Angular distributions from inelastic scattering on %: - - - (HF) and . . . (CC,), with coupling sum of HF and CC, ; x x x, CC, with coupling of at1 levels in fig. 8a only to the ground as iig. 8a; -, state; - . , -, sum of HF and CC,; W,(E) = 0.6E; data A taken from ref. 13).

Despite these different coupling modes in the ground-state deformation the angular distributions to the first 2+ states of these three nuclei show some similarity in shape for the same bombarding energy, as shown in figs. 5-7. This result is in agreement with that obtained at E, = 14.1 MeV [ref. ‘*)I and shows. that the interaction iS mainly located in the surface of the nucleus. The shape of the angular

M. Adel-Fawzy

et al. / 24Mg ) **si,

I I I I I I I I

I I I I I I I I 7 MeV

100

=S(n, n’)

47

I I I I I II

10 MeV

I-

1L MeV * St&on et al

10

LlOO VI \

8 MeV

11 MeV

E 0

10

-z au 32

-

100

10

--a-

FI

I LO

Fig. 7. Angular

I

I 60

I

I 120

distributions

I

(n,n,)

CC+ HF ........._.... cc

@Jdeg]

1

S

H/Z

II 160

from inelastic

scattering

on %;

data n taken

from ref. 14).

distribution is determined essentially by the potential radius (see table 2) and the transferred angular momentum at a given energy, if the direct excitation dominates. Similar results could be obtained also at measurements on selenium (at &= 6-10 MeV) [ref. 8)], tin (at E,,= 11 MeV) [ref. ‘)I, and samarium (at &,= 11 MeV) [ref. 4.5)] isotopes. The calculations of the angular distributions to the first 2+ levels were carried out using the coupling schemes shown in fig. 8a. In the case of 24Mg, the 6+ state is coupled only to the 4+ state in order to remain within the limit of the code CHUCK. This should not influence essentially the results because of the weak excitation of the 6+ state [see the magnitude of the related B( EA) values 46)]. A similar reasoning should be noted concerning the 3+ state coupled to the ground-state rotational band

48

M. Adel-Fawzy

in *%i. The coupling

in 32S includes

et al. 1 24Mg, “Si, 32S( n, n’)

commonly

only the first one- and two-phonon

states [see ref. “)I. In the reaction absorptive

model used here for description

part of the optical

potential

of the excited

has to be reduced

nuclear

depending

states, the

on the number

and excitation strength of the states which are explicitly taken into consideration. As a first step, the calculation for 32S was done without changing the absorption term W, = W,(SOM) = 8.50 MeV. As seen in fig. 7, the description of all experimental angular distributions by these calculations is quite good. All parameters used are listed in table 2; no energy dependence had to be introduced. The relatively smaller value of p2 = 0.30 for 32S [0.32 in ref. I’), 0.24 in ref. “‘)I is not expected to result in such a strong coupling as in 24Mg and 28Si. As a consequence, the absolute partial direct-excitation cross section for the first 2+ state in “S is smaller by a factor of approximately 2 than for 24Mg and **Si (see fig. 10). In the case of 24Mg, the absorption term had to be reduced to W, = 0.7 W,(SOM) = 5.43 MeV, due to the strong coupling of the ground-state rotational band O+-2+-4+6+ with p2 = 0.55 and p4 = -0.05 (see fig. 8a). Fig. 5 shows a comparison of these calculations with the experimental angular distributions; shape and magnitude are reproduced quite well. Also in this case no energy dependence of the optical parameters is necessary for a reasonable agreement with experiment. We now discuss in detail how one has to fix the absorptive potential W,. First, arguments for a general reduction of W, going from the optical model to the

p’,“’ = 0.55

p’,“’ = 0.48

,yt

p;“’ = - 0.3

= - 0.05

--p;”

p;” zo.2

= 0.2

p;‘b:0_2 8.”

t

6+ 6.28

4.12

1.37 00

iE 4+

2+

1.78

0’ 0.0

24Mg

+ +

4.62

T

-

32

% 2’

o+

24Mg

S

(bl Fig. 8. Coupling schemes tively; (b) K” =2+ band

used in the calculations: (a) ground-state in 24Mg; -, coupling scheme I; couplings, scheme III.

band and

and low-lying . . ., coupling

states, respecscheme II; all

M. Adel-Fawzy et al. 1 24Mg, 28Si, “S(n,

coupled-channel

method

are given above. The degree of reduction

number and type of states included and CC models require, in principle, bombarding

n’)

49

depends

on the

in the calculation. Furthermore, both the HF an energy dependence on W, increasing with

energy. In the partial statistical

model the absorption

term W, simulates

reaction channels which are not explicitly taken into account, and therefore it increases with incident energy due to the opening of more and more reaction channels. W, has the same physical meaning within the generalized optical model used for the CC method. Here, W, should compensate the restriction to few inelastic channels included in the coupled-channel calculation; the coupling scheme contains mostly a small part of open inelastic reaction channels, especially with respect to the increasing bombarding energy. Thus, the same meaning of W, in different models which are superimposed incoherently is a further reason to reduce this quantity and to choose it to be energy-dependent. Following these arguments the energy dependence for 24Mg was chosen simply to be linearly energy-dependent, W,(E) = aE with a = 0.5 MeV-‘. Using this absorption term within the HF model as well as the CC method, the angular distributions to the first 2+ state in 24Mg can also be described quite well as shown in fig. 9 (cf. fig. 5). This means the value of W, cannot be fixed definitely with sufficient accuracy in the case of analysis of the first 2+ state only, because the changes of the HF and CC parts due to variation of the quantity W, compensate each other. This is due to the fact that the first 2+ state represents the main part of the total direct inelastic interaction and so the consideration of the other channels (or not) is without remarkable influence. If this is true, one can expect a higher sensitivity of the energy dependence of W, for higher-lying states with weak collective nature, as will be demonstrated in detail in ref. 14). Using now the energy-dependent

absorption

term W, in the following

way (E in

Me% 32 s:

24Mg: 28Si:

W, = 8.50 MeV, w,=

OSE

forEG11

5.43 MeV

forEa12MeV,

MeV

W, = OhE,

fig. 10 shows in connection with the angle-integrated cross sections the competition of both components in the reaction mechanism at all bombarding energies investigated. In general, the direct excitation of the first 2+ states in these three even-even nuclei does not change rapidly with a bombarding energy between 7 and 14 MeV. The small differences in the curves - rising curve for 32S, decreasing curve for 28Si, and intermediate behaviour for 24Mg - are caused by the different energy dependences of W, only. The compound-nucleus contribution decreases rapdily with increasing energy due to the growing number of open channels for the compoundnucleus decay. In this way the direct excitation dominates at bombarding energies

50

M. Adel-Fawz~ et al, /’ “&fg, “Si, 32~(n, n’) 1

I

I

I

I

I

I

I

I

1

I

I

1

I

f

I

I

IOMeV

7MeV

100 -

I

i

I

I

I

m Stelson

.. . .

10 - ’ ----..

a...... ..+..,

--

--+4C

\ \

I

I 40

I

1

80

I

I 120

I

1 160

et al.

‘I-

....

_‘----.

l-

I

.y

..

,--7

I

14 MeV

II

I

10

0

-4)

II

/---.-

I

80

120

I

I

I_ 160

*cM!degl

Fig. 9. Angular distributions from inelastic scattering on 24Mg; W,(E) = 0.5E s 5.43 MeV.

higher than 10 MeV. The energy point where direct- and compound-reaction cross sections are equal is shifted down from about 9 MeV to less than 7 MeV for 32S and “Si, respectively. It can be demonstrated (figs. 5 and 9) that the first 2+ state in 24Mg can be described quite well using a constant term W, as well as an energy-dependent one.

M. AdeGFawzy

51

et al. / “‘Mg, *‘si, 3Zs(n, n’)

E

32S

0

l

d

'\

1000 2LMg

.

'\

--b* -<.---_

-E 100

-

L.

0

"A.,

V

----___-_ \

c

‘\ ‘\ \ \ ‘\ ‘\

<

\

1. l\ ‘\

‘A. 1.

10 -

l/I"""" 7 8

9

10

11

12

1.

13

1.

_ \-

1L

E,/ MeV sum of cross sections to the first 2+ states: - . - . -, (HF); - - -, (CC); -, Fig. 10. Angle-integrated HF and CC (see also text); data n , 0, A taken from ref. “), ref. 14) and ref. 13), respectively.

This means that from the investigation of the first 2+ state only we cannot conclude anything definite on the energy dependence of W,, and therefore the description of higher-lying states has to be included. The energy dependence of W, is stated above from a physical point of view, i.e. from the underlying model picture. But other model parameters are also known to be energy-dependent, as for instance the real potential well depth V,. In this analysis the following method is chosen. From analyses of the elastic nucleon scattering in a wide energy region it follows that the real potential well depth must be chosen to be energy-dependent: V,(E) =

M. Adel-Fawzy

52

Vr’-

et al. / 24Mg, 28Si, 3ZS(n, n’)

(YE [see, for instance,

refs. 42,44)], whereby (Y is in the order of 0.2-0.5. This is also used for description of the inelastic scattering 3*4), addi-

energy dependence tionally

of W,(E).

to an energy dependence

Because

a fit search code such as ECIS

was not available, we took parameters from the literature and varied them in large steps. By comparison with the exceptional angular distributions and interpolation the final parameter set was fixed. In order to test the role of the energy dependence of V,(E), in the case of 28Si calculations were carried out for V,(E) = 52.0 MeV= const and V,(E) = 56.0 MeV- 0.3E at incident energies E. = 7 and 14 MeV, respectively. The deviations in the angular distributions to the ground state as well as to the first 2+ state following from the CC calculations are not remarkable [see ref. ‘“)I; the differences of the angle-integrated cross sections for these two cases are 3.0% (0:) and 6.2% (2:) at Eo= 7 MeV and 0.3% (0:) and 4.4% (2:) at Eo= 14 MeV, respectively. We concluded, that the energy dependence of W, is dominant, and V, was kept constant. It follows that the energy dependence of W,(E) = aE may reflect an averaged behaviour only in the energy region investigated here. At lower incident energies another dependence may give a better description 4). Let us now consider how to fix the energy dependence of W,. In the case of 32S this was impossible, because only the first 2+ state could be experimentally resolved (see also fig. 3). We will discuss this effect for the case of 24Mg, because for 28Si similar results could be obtained, as outlined in detail elsewhere 24). Fig. 11 shows that for 24Mg the value W,= 5.43 MeV (= 0.7 W,(SOM)) gives too high HF cross sections

at bombarding

up to 9 MeV. At E. = lo- 14 MeV the HF contribu-

energies

2’ 0= -4.1228MeV

1"=4+

0 = -4.2385MeV

I" =2'

I -

I

1

I

I

I

I

Mg

(n,n’)

0= -5.236MeV 1"=3* I

I

I

I

I

Q =-6.01 MeV I"= 4+ I

I

I

I

1

I

l

+\

4d+

100

\0

z E

0

m--

\

\

0

-=y$

4

\4

\

10

\ I

1

8

I

1

10

I

1

12

I

:i I

I

14

8

1

I

10

I

0 --co __ \ 0

4 6-

\‘E

\_ \I

I

I

12

8

1

1

10

I

I

12

E, [MeVI Fig. Il.

Angle-integrated

cross sections

from the 24Mg(n, n’) reactions; with W,(E) = const.

the curves are HF calculations

M. Adel-Fawzy et al. / 24Mg, “Si, “S(n,

tion becomes

essentially

smaller than the experimental

points. Therefore,

energy dependence for W, = 0.5E c 5.43 MeV seems to be reasonable, limit of 5.43 MeV (= W,( E = 11 M&V)) can be understood. Our results

concerning

the excitation

53

n’)

the chosen

and the upper

of the first 2+ state in 24Mg (similar

for ‘*Si

and 32S) can be summarized as follows: (i) The optical potential parameters were chosen individually from the literature and selected by comparison with our elastic scattering data at all bombarding energies from 7 to 14 MeV. (ii) HF calculations were done with the reduced W, =0.7 W,(SOM) using all existing ihdividual levels for the (n, n), (n, p) and (n, a) channels at bombarding energies 7, 8 and 9 MeV. For higher incident energies level continua in the different residual nuclei were included. (iii) The direct-reaction contributions following from the coupled-channel calculations with coupling within the ground-state rotational band and with the same optical potential parameters as for the HF calculations were added. (iv) A constant W, value resulted in too high an HF contribution for the high-lying levels. From this point of view and also in agreement with general physical aspects, W, was chosen to be energy-dependent. This results in proper HF contributions for the high-lying states at the lowest bombarding energies discussed here. (v) The energy-dependent W, introduced into the HF calculations as well as into the coupled-channel ones also gives a good description for the 2: angular distributions. It is quite clear that a consistent description of all considered levels at all bombarding energies investigated here requires that the same parameters in both underlying models have to be used. This is achieved by introduction of the energy dependence of W,, whereby the function W,(E) was found empirically and may depend on the energy region studied.

4. Excitation

of the other states in %Mg

The low-lying states in 24Mg can be described by the following structure model which underlies the analyses in this work. The I “/ E, states O+/O.O, 2+/ 1.37,4+/4.12, 6+/8.11... are members of the K w = O+ ground-state rotational band. The energy ratios Ed/E2= 3.01 (3.33) and E6/E4= 1.97 (2.10) deviate less than 10% from the theoretical values (in brackets) following from the first-order semiclassical E,= h21(1 + 1)/25 rule. This indicates that other structural components, i.e. -y-rotations, vibrations or others are expected to be small for these states. This structure model is also used in different investigations of inelastic scattering 1’,27*28*42) but follows from B(E2) transitions 47) and measurements of the electric quadrupole moment 48-5o) for the ground-state rotational band. The states 2+/4.24, 3+/5.24, 4+/6.01 . . . can be understood as members of a K n = 2+ band. In this case a deformation perpendicular to the symmetry axis is

54

M. Adel-Fawzy et al. / 24Mg, **Si, ‘*S( n, n’)

possible leading to a dynamic non-axial deformation 5’). Another possibility is the description of a statically, non-axially deformed rotator 38,52).Both types of deformation should

be equally

of the collective

model.

adequate

in the adiabatic

The state 0+/6.43

approximation

is the head of a second

within

the frame

KT = O+ band.

This state was resolved experimentally only at the incident energy E, = 9.0 MeV. In fig. 13 it can be seen that the compound-reaction contribution consistently calculated as described above cannot reproduce the experimental cross sections in the full energy range investigated. It follows that direct-reaction contributions have to be taken into account. In the present work the CC calculations in the energy range 7 to 14 MeV were carried out separately for the K ?i = O+ and Km = 2+ bands. This means that coupling (mixing) within these two bands is not taken into account because of the limits in the code CHUCK, excluding the ground state which represents the initial state [see also ref. “)I. Following the structure model cited above the states of the ground-state band are described as pure rotational states with deformation parameters according to table 2. The 2+/ 1.37 state is discussed in detail in sect. 3. The elastic scattering has been analysed in the frame of the SOM as well as the CC representation (see fig. 4). The coupling to the ground-state band shows only a small influence on both the shape and magnitude of the angular distributions, indicating that the reduction of the absorptive term W, in the CC calculations reflects the explicit consideration of the “main” reaction channels in an adequate manner. Further conclusions are impossible because of the relatively small number of experimental data points. The 4+/4.12 state could not be resolved experimentally from the 2+/4.24 one. Therefore, the calculations to this 4+ state were done according to the coupling scheme in fig. 8a. Results are shown in fig. 12. These results should be considered in connection with the interpretation of the K r = 2+ band. It can be seen that both levels of this doublet contribute in the same order of magnitude to the experimental observed cross section. The Km = 2+ band is interpreted in the present work as a rotational band based on a quadrupole vibration of the prolate ground state. The bandhead is the 2+/4.24 state. The excitation energy for members of such a K n band in first order should follow the &(I”,

K) = EK +h’I(Z+

1)/2J

rule. In our case the energy

ratio (Ed-

E2)/(E3 - E2) = 1.78 shows a deviation of less than 25% from the theoretical value of 2.33. In other works 38.42) this structure model has also been used with success. Furthermore, the inner quadrupole moment of the K H= 2+ band is higher than for the g.s. band 46). The deformation parameter pi for this K TT= 2+ band can be derived from the p2 value related to the ground-state band according to pi = /32(J2/J0)“2, where the ratio of the momenta of inertia can be taken from the energy spacing within these bands. In our case this ratio leads to the value p;=O.7. The first analysis starts with the coupling scheme I according to fig. 8a. The results for the 2+/4.24 state are shown in fig. 12, together with the 4+/4.12 calculations to the non-resolved doublet. The sum of the different contributions in principle gives the right order of magnitude showing the relevance of the underlying structure

h4. Adei-Fawzy

. ..-.

7 MeV

l-

et al. / 24h4g. “Si, 32S( n, n’)

55

10 MeV

+

f

__-.w.--

-“...._.(‘-.

I..

. . . . . . . . . . . . . . . . . . . . * **_t.....l*.l

. . . . . . * I...*_*

-

10

80

120

160

10

60

120

CC+HF

160

QcM[degJ

Fig. I?.. Angular distributions from inelastic scattering on 24Mg to the unresoked doublet; CC(ly”), coupling as fig. 8a; CCf2+), coupling I as fig. 8b; CC,(2+), coupling II as fig. 8b.

56

M. Adel-Fawzy

et al. / 24Mg, *‘Si, 32S(n, n’)

21Mg (n,n’) Q=-5.236MeV

I

I

In=3*

I

I

Q=-6.01MeV

I

I

I

I'=&'

I

I

I

100 -

.-

-*-

_.C.

___./*

l1

1

0

I

10

I

I

I

12

8

I

I

I

10

I

12

E,tMeVl Fig. 13. Angle-integrated

cross sections from the 24Mg(n, n’) reactions: with coupling I as fig. 8b; -, sum.

- - - (HF)

and -

-

- (CC),

model and model parameters chosen. In other words, fig. 12 does not show a remarkable contradiction with the uniform HF calculations outlined in sect. 3 or with the direct components following from the CC calculations. Another conclusion is, that for the higher-lying members of this K n = 2’ band the cross sections of the direct contribution calculated in this way are too small, as demonstrated

in fig. 13, especially

at higher bombarding

convincing arguments for the consistency of the If we take only ELIESE results into account, calculations for the open (n, n), (n, p) and (n, a) the experimental points in fig. 13 are reproduced

energies.

This figure gives

HF calculations described above. i.e. 30 discrete levels in the HF channels without level continuum, approximately. But we know that

this picture is rather unphysical; the number of open channels at E,, = 12 MeV is much greater than at E,= 9 MeV. Therefore, we cannot simply simulate another amount of HF contributions as presented here; the existence of some direct reaction contributions is evident. As a competition to the excitation via the K r = 2+ rotational band, the direct excitation from the ground state was taken into consideration. In agreement with the known electromagnetic transition probabilities B( EA) [ref. “‘)I a direct coupling of the 4+/6.01 state with the ground state was included into the CC calculations (see coupling II in fig. 8b). The results are shown in fig. 14. The direct one-step excitation is of order of 75% in competition with the multi-step excitation via the K Ti= 2+ band. Similar results were obtained

in a (p, p’) interpretation

42). This means

M. Adel-Fawzy et

I

I

I

I

I

I

I

I

nl. /

III

24Mg, *‘Si, 32S( n, n’)

57 I

II

III

I

LO

I

I

80

I

I

120

I

I

160

CC+HF --- HF .....-*CC --x-Cccl

LO

120

80

160

LO

80

120

160

ocMkkgl Fig. 14. Angular

distribution

from inelastic

scattering

on “‘Mg: CC, coupling

II as fig. gb;

CC,, coupling

III as fig. Sb.

in a simple surface

picture

vibration

that an essential in comparison

part of excitation

with the rotational

energy is concentrated energy.

in the

On the other hand, one

cannot expect that a simplified model is able to reproduce all the fine structure of the angular distributions. In this sense, fig. 14 demonstrates an averaged and rather crude result for description of such a high-lying state investigated for the first time in neutron scattering. For excitation of the 3+/5.24 state similar arguments should be valid. But for this non-normal-parity state information from electromagnetic transitions has not been .I. published. Nevertheless, an additional coupling to the ground state was also introduced corresponding to a one-step spin-flip process, as shown in the coupling scheme III in fig. 8b. This description gives an enhancement of the direct contribution by a factor of about two and also a proper result (see fig. 15), showing that the vibrational excitation from the ground state of all members within the K V = 2+ band has consistently the same strength.

58

hf. Adel-Fawzy et al. / “‘Mg, 28Si, 32S(n, n’)

*4Mg (n,nL 1 I

6

I

O=I

8

I

5236MeV

I

I

10

I

12

1'=3+

1 1

1L

E, IMeVl Fig. 15. Angle-integrated cross sections from the 24Mg(n, na) reaction: - - - (HF) and - . - . - (CC,), with coupling I1 as fig. 8b: -, sum of HF and CC,: - . . -, CC2with coupling III as fig. 8b; - . . . -, sum of HF and CC,; data l taken from ref. 54).

A similar model was used for interpretation of the excitation of the 3’ state in (p, p’) experiments in the energy range from E,= 17 to 35 MeV [ref. 42)], but good agreement with the experiment was obtained only at lower bombarding energies. It should be noted that the phase conditions between the two components AS = 1 and CC (K TT= 2’ band) are not known and therefore a simple addition of these contributions is questionable. But this simple picture gives the right order of magnitude for the integrated cross sections, and for the differential angular distributions in a similar manner to fig. 14. Additionally, in fig. 14 can be seen the difference between coupling schemes II and III with respect to the description of the 4+/6.01 state. This difference is negiigible, and so the final result for all states of the K” = 2+ band is represented uniformly by the coupling scheme III which includes the direct excitation from the ground state to all states investigated. 5. Summary

Using a time-of-flight multi-angle detector system, investigations of neutron scattering to low-lying states of the light even-even nuclei 24Mg, 28Siand 32S have been performed. The investigated energy range between 7 and 14 MeV is of special interest because of the quantitative description of the competition between the compoundnucleus mechanism and direct interaction. The simultaneous measurement of higherlying states in neutron scattering provides information that has been missed in earlier

M. Adel-Fawzy

works. In this way a voluminous

et al. / 24Mg, 28Si, “S(n,

and consistent

quantity

59

n’)

of experimental

material

is presented. The analyses were performed in the frame of incoherent superposition of a compound-reaction part (Hauser-Feshbach model) and a direct contribution following from the collective model in the coupled-channel representation. The influence of both models on each other is concentrated on a modification of the absorption term in the optical potential and has to be chosen depending on incident energy. The other optical-model parameters were taken from published works and checked using the measured elastic scattering differential cross sections. The deformation parameters were also taken from literature and tested by comparison with the first 2+ angular distributions. The nuclei investigated show a typical rotational structure, vibrational excitation spectrum, and intermediate behaviour for 24Mg, 32S and %, respectively. Also the averaged nuclear field is different, as expressed by the difference in optical parameters. Nevertheless, there are some similarities in the shape of the angular distributions for the first 2+ states of all three nulcei. The description of these angular distributions in shape and magnitude is quite excellent because the coupling to the ground state is the absolutely dominating process. The competition of the compoundand direct-reaction mechanisms can be determined qualitatively: the direct-interaction contribution only weakly depends on incident energy, whereas the compoundnucleus contribution decreases rapidly with increasing bombarding energy. The exact energy dependence of both components is directly related to the energy dependence of the absorption potential W,(E). The latter can be fixed more definitely only by taking the higher-lying states into consideration. The low-lying states in 24Mg can be classified into the K r = Ot ground-state rotational band and Km = 2+ band, respectively. The latter can be understood as a rotational band based on a quadrupole vibration of the deformed ground state. This vibrational excitation leads to a direct coupling of the K 71= 2+ band members to the ground state. Within this nulcear structure model all experimental angular distributions could be described consistently. The higher-lying states were measured and interpreted for the first time. For these states it is felt to be Sufficient if the averaged magnitude of the different cross sections is reproduced. The fine structure in the angular distributions cannot be described with the computer codes used. This may be due to the simplification within the nulcear structure model as well as some neglection of the excitation (coupling scheme) which was required by the limit on computational expence.

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n’)

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n’)

61

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