tt On leave of absence from Instituto de Fisica da Universidade de Sao Paula, Brazil. ttt On leave of absence from Pontificia Universidade Catolica do Rio de ...
Nuclear Physics A238 (1975) 120-M; Not to
@ North-Holland Publishing Co., Amsterdam
be reproduced by photoprint or microfilm without written permission from the publisher
EVIDENCE FOR CRlTICAL ANGULAR MOMENTA IN THE FORMATION OF 26A1 VIA THE “N+“C C. VOLANT,
M. CONJEAUD,
S. HARAR,
CHANNEL
S. M. LEE 7, A. LGPINE tt
and E. F. DA SILVEIRAttt Service de Physique Nuclt?aire ci basse tkergie, CEN Saclay, BP2, 91190 Gif-sur-Yvette, France Received 8 August 1974 (Revised 11 October 1974) Abstract: States in *ONe have been studied through the 12C(14N, 6Li)20Ne reaction. Excitation functions have been measured from 20 MeV to 60 MeV in steps of 5 MeV at different angles for *ONe states up to 10 MeV excitation energy. States of 24Mg have been also populated using the 12C(14N, d)24Mg reaction; excitation functions of 24Mg states up to 9 MeV excitation energies as well as angular distributions at 35 MeV bombarding energy have been obtained. Comparisons of data with Hauser-Feshbach calculations show clearly that the compound nucleus mechanism is the main process for both ‘*C(14N, 6Li)20Ne and 12C(14N, d)24Mg reactions. Strong evidence has been provided for inhibition of the 26A.l compound nucleus formation for angular momenta higher than critical values. The location of the yrast line in the 26A1 nucleus is discussed.
E
NUCLEAR REACTIONS 12C(14N, 6Li), E = 20-60 MeV, measured a(E; 0); 12C(14N, d), E = 20-40 MeV, measured a(E); E = 35 MeV, measured o(0); Z”Ne and 24Mg deduced levels. Hauser-Feshbach calculations and suggested critical angular momenta for the 26A1 compound nucleus formation.
1. Intrdduction The role of critical angular momenta (J_) in heavy-ion reactions has been extensively studied by measuring complete fusion cross sections ‘). In this work, it will be shown that the J,, values can be derived for a wide range of excitation energy in the compound nucleus from the Hauser-Feshbach (HF) analysis of individual level cross sections in appropriate outgoing channels. Indeed, recently, Hanson et al. “) found that the statistical model fits rather well the angular distributions and the absolute cross sections of levels observed in the 12C(14N, 6Li)20Ne reaction studied at 52 MeV and at 76 MeV by Marquardt et al. and by Belote et al. respectively “). They also pointed out the necessity to introduce in the HF analysis a critical angular momentum lower than the grazing angular momentum (J,,) involved in the 14N+ “C channel. But, such a determination of 7 Also at Institut fiir Kernphysik der Universitiit zu Ktiln. tt On leave of absence from Instituto de Fisica da Universidade de Sao Paula, Brazil. ttt On leave of absence from Pontificia Universidade Catolica do Rio de Janeiro, Brazil. 120
CRITICAL
ANGULAR
121
MOMENTA
+o
+z
E91
00
:
_; c
122
C. VOLANT et al.
Jcr is strongly dependent of all kind of parameters used in the HF formula, and principally of the level density parameters for some outgoing channels. Moreover, uncertainties in the absolute experimental cross sections might be also important; for example Hanson et ~1.“) obtained different normalization coefficients to fit the experiments at 52 MeV and 76 MeV. The aim of the present work is to clarify the role of Jo in the 26A1 compound nucleus formation. For this purpose we studied the ‘2C(‘4N, 6Li)20Ne reaction in a wide range of incident energies, namely from 20 MeV to 60 MeV. We also extended our study to the 12C(14N, d)24Mg reaction because we found that this channel is rather insensitive to Jcr and then, its analysis heIps to fix many parameters in the HF formula. The physical meaning of Jcr values, as well as the sensitivity of the present method to determine them, will be discussed.
2. ~x~rimenta1 procedure and results The 12C(14N, 6Li)20Ne reaction has been studied from 20 to 60 MeV incident energy by using the ’ 4N5 +I 6 + beams from the FN Tandem Van de Graaff of Saclay. Detection and identification of particles were achieved by three conventional E x dE counter telescopes. Carbon targets were about 20 pg/cm2 thick and had a thin deposit of gold (around 1 pg/cm’). To obtain absolute values of cross sections, the integrated beam charge was monitored by the Rutherford elastic scattering of nitrogen projectiles on gold at forward angles; 12C target thicknesses were measured by studying
EXCIT,%;ION ENERGY IN *%g
r-
(Me!/)
16 ~__&__
Ii0
+_
200
250
300
350 CHANNEL
400
450
500
550
600
650
NUMBER
Fig. 2. A deuteron energy spectrum from the 12C(14N, d)24Mg reaction at 35 MeV incident energy and 7” in the laboratory system.
CRITICAL
ANGULAR
123
MOMENTA
the energy loss of alpha particles through them. In order to minimize carbon built up on the targets, they were surrounded by a piece of copper cooled by liquid nitrogen. Absolute cross sections are believed to bs accurate within 40 %. They are in good agreement with data obtained by Nagatani et al. “) at 60 MeV, but values published by Marquardt et al. “) are systematically three times smaller than the present data. A typical spectrum recorded at 60 MeV is shown in fig. 1. The overall energy resolution was about 200 keV (FWHM). In view of the rather smooth angular distributions at 52 MeV and 76 MeV, we restricted our measurements to three angles: 15”, 22.5” and 30” in the laboratory system. Excitation functions were studied by steps of 5 MeV since the energy average behaviour of cross sections was shown to be smooth by Marquardt et al. “). The 12C(14N, d)24Mg reaction has been studied between 20 and 40 MeV incident energies at 15” in the laboratory system. Absolute cross sections are in good agreement with those obtained at 20 MeV by Erb et al. “). A spectrum obtained at 35 MeV is presented in fig. 2. The energy resolution is about 120 keV (FWHM). Angular distributions have been measured at 35 MeV between lo” and 170” in the c.m. system. 3. Hauser-Feshbach calculations 3.1. FORMALISM
AND
CHOICE
OF PARAMETERS
We have analyzed our data using the Hauser-Feshbach the averaged compound cross section has the form
theory; in this formalism
The summation over J rules out from 0 to co in principle. In eq. (1) unprimed and primed indices refer to the entrance and outgoing channel respectively. The quantities a” indicate all possible exit channels decaying from the compound nucleus. The notations for the quantum numbers have the following meaning: i: spin of the projectile; I: spin of the target nucleus; S: the channel spin s = I+ i; 1: the orbital angular momentum; Jn: the spin and parity of the compound state, 3 = Z+S = l’+S’ = l”+S” 7Z= 7c,(- 1)’ = 7&J- 1)” = n,.,( - 1)“‘. The expression for the differential cross section differs from formula (1) only in the introduction of geometrical factors and is not shown here. The code used has been described by Berthier et al. 6), when they performed the analysis of the 12C(160, U) 24Mg reaction. Optical model parameters used to calculate the transmission coefficients TF for all channels considered in the HF analysis are presented in table 1.
3.85 ‘)
(fin)
3.36
10.8 4.33 0.58
90.15 4.33 0.58
3.71
6.79
6.58
‘) Ref. 20).
‘) Ref. II).
1.35 (‘41.)+&j) 0.45
3.87 1.48
5.92 0.26
1.35 (Al*+&+)
0.4+0.125 EC.,,,.
0.41 12.0
7.5+0.4 Ecam. 1.35 (k-f+&+) 0.45
6.64 111.7
“1
l*F+*Be
4.02
3.41 ‘) 2.25 0.15
‘1
*‘NefSLi
65.5
5
3.84 b, 2.67 0.11
‘)
23Na+3He
5.60 0.48 27.0
25
10.4 5.35
3
20Ne+6Li
100
d, Ref. 9).
0.59 4.33
0.59 16.0 4.33
0.70 11.0
0.66 6.6 3.50 0.66 3.65 0.7
50 4.33
40 3.65
53 3.71
Imaginary wells are of the volume type. Optical parameters from: ‘) Ref. 19). 3 Ref. lo). b, Ref. 12). Level density parameters from: “) Ref. 14). ‘) See text. *) Ref. Is).
RC
a~ (fm)
4 (fm)
R (fm) a (fm) W (MeV)
V (MeV)
5
11
0.13
5.13
3.58 h,
‘)
14N+‘2C
discrete levels
0.15
3.85 ‘) 2.67
7
b,
10
2.46 0.15
24Mg+d
25Al+n
9
3.67 E, 0 0.12
7
9
EC(MW No of
Y
A ‘1
a
=Mg+p
22Na+a
TABLE 1 Level density parameters and optical model parameters
9.84 15.16
‘1
160+10B
s
%
3 2
CRITICAL
ANGULAR
MOMENTA
125
The denominator of the HF formula (1) includes all decay modes energetically open from the compound nucleus, G(J) = ( c
l;:“)Jz.
(2)
a”l”S”
Quantities (2) were evaluated for outgoing channels by using discrete levels of known
50
k
5.63 + 5.80 3I-
/e.--_
20
++*
20
'\
7.$+7.17+7.*0 + 3’ o+ loo /---. t’ t 50 [-\
10
1 200
7.43
1W
2+
\
\ \
\ j
\
\
yg-&
'4
10
5
!-\
10
/
/...... .....*......*..
200 100 50
4.25 4+
;1‘-++*, *.....** '..,..,.,
9 -\ tP*tt+\+ t-y
.
\
\
1 10 L
4.97 220 10
\ \
I
20
f--N
I 30
I LO
I 50
E$(MeV)
I 60
I 70
/ T t
_!-
9.04+ 9.12 4+ 3-
c-
50 E
:
!
5i
I-
\
+
-II -
30
I
I
I
LO
50
60
I 70
z
“N hb(MeV)
Fig. 3. Excitation functions for some ‘ONe states measured at 15” in the laboratory system for the 12C(14N, 6Li)20Ne reaction. Levels are labelled by their excitation energy, spin and parity. Curves (1) (dotted line), (2) (dashed line) and (3) (solid line) are from HF calculations without any normalization (see text for definitions).
126
C. VOLANT et al. 12C(“N ,%i ) 2sNe
8[ob=22’5 7.0 + 7.17+ 7.20 L- 3o+
9,s
Fig. 4. Excitation
functions for some zONe states measured at 22.5” in the laboratory the 1zC(14N, 6Li)20Ne reaction (see caption to fig. 3).
system for
spins and parities up to EC (table 1) and then by using level densities,
where The where The
0 S I” < 00 and IJ-Z”l 5 I” 5 J+Z”. cut-off of the yrast line was made by using the relation Err= YZ"(Z"+1) Y = h2/2~~i, where Sri, is the rigid body moment of inertia; Eli, = # A’. level density formulas used were P(& Z) =
1
e24zF(z)
12a*(2a*)+ US’4
’
(4)
where F(Z) = (2Z+ l)e-I(‘+ 1)‘2a2.
(5)
CRITICAL
ANGULAR
MOME~A
127
‘LN
ELob PW Fig. 5. Excitation
functions for some z”Ne states measured at 30” in the laboratory 12C(14N, 6Li)20Ne reaction (see caption to fig. 3).
system for the
The quantity U = E-A = at2 is the excitation energy E corrected for the pairing energy A; t is the nuclear temperature, a is the single-particle level density parameter and o2 = ~~i*tlh’ is the spin cut-off parameter. The level density parameters are listed in table 1. Although generally one calculates the denominator G(J) by using known discrete levels at low excitation energies and the level density formula for the, continuum energy region, there has been found little difference from using only the level density formula for the whole excitation energy. 3.2. COMPARISON
WITH EXPERIMENTAL
DATA
Excitation functions of “Ne levels up to 10 MeV excitation energy measured at different angles are presented in figs. 3, 4 and 5. Higher lying “Ne states are not shown since the present energy resolution does not allow a reliable comparison with
C. VOLANT et al.
128
TABLE 2 Values of JSr and J,, for different incident energies
Incident energy (MW
Compound nucleus excitation energy (MeV)
20 25 30 35
24.3 26.6 28.9 31.2
40 45 50 52 55
33.5 35.8 38.2 39.0 40.4 42.8 50.2
60 76
.I (6
7 9 11 13 14 15 17 17 18 19 21
J
J/4,*,
(fii
(11) 12 12 13 14 14 15 16 17
0.69 0.71 0.67 0.73 0.79 0.77 0.85 0.89 0.89
The excitation energies for 26A1 are also indicated. The last column is a comparison between the deduced moments of inertia and the rigid body value J,,, = $roZA$ where r0 = 1.25 fm (X/4,,, = 0.51 for the g.s. band of 26A1).
HF predictions. Curves labelled (I), (2), (3) sh own in fig. 3 correspond to different HF calculations. The dotted line (1) is obtained by assuming that angular momenta up to the Jgr values in the entrance channel contribute to the 26A1 compound nucleus formation (J,, is choosen equal to the angular momentum for which the transmission coefficient is 0.5). This curve fits quite well the few experimental absolute cross sections obtained below 30 MeV incident energy; but the discrepancy between theory and experiment is strong at high incident energies. This disagreement is more pronounced if expansion (1) is performed up to J > Jgr. Then, to improve the fit, it is necessary to choose a critical angular momentum J,, lower than Jgr at incident energies higher than 30 MeV (see table 2); the results of these calculations are the solid curves labelled (3). To study the sensitivity of the HF cross sections to the Jcr values, we performed another calculation assuming J,,+ lh (where J,, are those shown in table 2) for the limit of expansion (1); the results are illustrated by curves labelled (2). One observes that the HF cross sections are strongly affected by the limit of Jcr at high incident energies, and to a lesser extent at low energies; in particular we performed a calculation assuming J,, + 2h at 25 MeV; changes in HF cross sections Of less than 20 % are obtained. It is interesting to notice that the cross section at 15” (fig. 3) of the 2+ state at 7.83 MeV is a factor of 2 larger than the HF predictions, on the contrary the cross section of the closed 2+ state at 7.43 MeV is well fitted. Ford et al. “) studying the log (160, 6Li)20Ne reaction measured equivalent cross sections for these two states which are moreover well fitted by HF predictions. Middleton et al. ‘) studying the
CRITICAL
ANGULAR
MOMENTA
129
“C(r2C, a)“Ne reaction showed that the 7.83 MeV state has an important 8p-4h configuration. It seems reasonable to assume that at least a small component of the “C ( 14N ’ 6Li)20Ne reaction mechanism proceeds via a direct eight-nucleon transfer and selects preferentially such states at forward angles. This effect becomes smaller at 22.5” and 30” and seems more important at incident energies lower than 50 MeV. To check further the reliability of the parameters we have used in the present HF calculations, we analyzed the data obtained by Marquardt et al. “) at 52 MeV. Results shown in fig. 6 fit quite well the shapes of angular distributions, but as already mentioned by Hanson et al. ‘) the HF absolute cross sections are 2-3 times larger than experimental values. So we measured the 12C(14N, 6Li)20Ne reaction at 52 MeV at a few angles and we found cross sections systematically larger by a factor of 3 than those published in ref. “); therefore, the present HF calculations fit rather well absolute cross sections without any arbitrary normalization.
‘*C (“N pLi )*‘Ne El& = 52MeV IOF.
’
I
I
I
I
j
7
E,=
II
I t
T
5.63+ 5.78
Fig. 6. Angular distributions for some 2oNe states measured at 52 MeV by Marquardt et al. 3). Curves are results of HF calculations divided by a factor of 2 (see text for this normalization).
130
C. VOLANT et al.
12c PN E,$
I
I
0
/
I
LO
80
I
120
l@J kll
6Li 12’Ne 76 MeV
I
I
0
_
-L-II
LO
80
’
120
1
160 Q,,;
I
Fig. 7. Angular distributions for some *ONe states measured at 76 MeV by Belote et af. ‘). Curves are results of HF calculations without any normalization.
We also analyzed angular distributions obtained by Belote et al. “) at 76 MeV; as found in ref. “) we obtained a fairly good agreement for both shapes and absolute cross sections (fig. 7). As we will discuss in the following section, the study of the 12C(14N, d)24Mg reaction and its analysis in the framework of the HF theory is a good test of the parameters involved in this model. Curves (1) and (3) in fig. 8 are obtained under the same conditions as those shown in fig. 3; but now they are quite similar, contrary to what it is observed for the r2C(r4N, 6Li)20Ne reaction at incident energies above 30 MeV; the relative and the absolute magnitudes of the cross sections are fairly well fitted at different incident energies. Our recent measurements of angular distributions of the “C(14N, d)24Mg at 35 MeV are shown in fig. 9. Dashed curves are HF predictions and fit well the shapes and the relative magnitudes of the cross sections, but the theoretical cross sections obtained with the parameters used in most of this analysis, underestimate the absolute magnitude by a factor of 1.5. This point will be discussed later. We notice that the O+ state at 6.44 MeV and the 2+ state at 7.35 MeV in 24Mg show a discrepancy with the HF predictions but present results are too fragmentary to speculate on the multiparticle-multihole nature of these states.
CRITICAL
ANGULAR
12’C(‘LN,
MOMENTA
131
d )2LMg
@L,b = 15’ 6.LL MeV Of
g.5 l
0+
t
*
*
+
\
t
lj \ :.,
t
t
\..
til \ 13
A.
L.12 + A.23 MeV 4+ 2+
DC
I
-0-o
100
1 -r
Et,b (MeV) Fig. 8. Excitation
Etab (MeV)
functions for some 24Mg states measured at 15” in the laboratory 12C(14N, d)24Mg reaction (see caption to fig. 3).
3.3. THE CRITICAL
ANGULAR
MOMENTUM
system for the
J,r
Now we will discuss the important role played by the critical angular momentum J,, in the 26A1 compound nucleus formation. We can expand the HF average cross section for an outgoing channel in terms of partial cross sections, m
i&
=
c .I=0
e(J).
(6)
132
C. VOLANT ef
‘*C(14N , d)
al.
24Mg
200-
-*--___---
ecm.
8 c.m
Fig. 9. Angular distributions for some 24Mg states measured at 35 MeV. Curves are results of HF calculations multiplied by a factor of 1.5 (see text for this normalization).
CRITICAL
ANGULAR
133
MOMENTA
When we have to introduce J,, in (6) for fitting the experimental cross sections, the sum (6) becomes
Expansion (6) is shown for some “Ne and 24Mg states in figs. 10 and Il. Few a(J) terms are predominant around the maximum value e,_(J). At 40 MeV, for example the o(J) expansions of “Ne states is p eaked around J = 14 h which is close to Jgr in the entrance channel (see table 2); but for the 24Mg states, the o(J) expansions are peaked at J-values smaller than J,,, namely between 8 h to 11 h. So it is clear why the 6Li+20Ne cross sections are more sensitive than the d+24Mg cross sections to the J cut-off in the a(J) expansions near Jgr. This is illustrated in figs. 3 and 8 by the comparison of curves (1) and (3). To study the role of Jcr in the 26Al compound nu cleus formation, it is important to correctly calculate the denominator G(J) which represents the total cross sections of all competing outgoing channels from the compound nucleus. Contributions to G(J) are shown in fig. 12. In the angular momentum range where the a(J) expansions of the 6Li + “Ne and d+ 24Mg channels are peaked (figs. 10 and 11) it is obvious that only few channels are important in G(J). These are the c(+ 22Na, p + 2‘Mg, n + “Al and d+24Mg channels. Their relative contributions to the d+24Mg and 6Li+20Ne cross sections are varying with the incident energy: above 30 MeV the u+ “Na
‘*C(“NfLi)
0
4
a
12
16
*‘Ne
2c
1
24 J,
Fig. 10. HF predictions for the a(J) expansion for some states of “ONe populated 12C(14N, 6Li)20Ne reaction at different incident energies.
by the
134
C. VOLANT et af.
12C(‘4N,d)24Mg EI~N
0.1 0
Fig. 11. HF predictions
= 40MeV
-L-l .C
8
12
16
20
22
Jr
for the a(J) expansion for some states of 24Mg populated 12C(14N, d)24Mg reaction at 40 MeV.
by the
channel is predominant but at lower energies the p+ 25Mg channel becomes important. Then it is necessary to correctly evaluate the parameters used for these channels. This choice can be checked by studying the 12C(14N, d)24Mg reaction at different incident energies. So we analyzed the “C(14N, d)24Mg reaction performed by Erb et al. “) at 20 MeV. As found by these authors the HF analysis fits quite well both the shapes of angular distribution and the absolute cross sections. At this incident energy the o(J) expansions for levels observed in the d+24Mg channel are peaked in the J-space where the p+25Mg channel is predominant for G(J); then the good fits obtained give confidence in both the optical model parameters and level density parameters of the p+ “Mg channel, but do not help to fix the parameters for the c1+ 22Na channel. The HF cross sections of the d+24Mg states observed above 30 MeV incident energies are strongly dependent on the behaviour of the a+ “Na component in G(J) and are not very sensitive to the J-limitation (see fig. 8). The comparison of HF predictions and experimental results in these particular situations constitutes a nice test for the choice of the level density parameters of the a+ 22Na channel. Once these values are fixed, the HF analysis of the 6Li + “Ne channel above 30 MeV allows to determine J_ at each incident energy in order to obtain the best fit to experimental results. These values are shown in table 2.
CRITICAL
ANGULAR
MOMENTA
Fig. 12. Contributions of open channels to the HF denominator 33.46 MeV in the 26A1 compound nucleus. (This corresponds 3.4. UNCERTAINTIES
13.5
G(J) at an excitation energy of to 40 MeV incident energy.)
IN HF CALCULATIONS
3.4.1. Transmission coef$cients. The optical parameters we used to &rive the transmission coefficients T, for the entrance and outgoing channels are shown in table 1. We will discuss now uncertainties due to ambiguities in these parameters. At incident energies lower than 40 MeV, HF cross sections are significantly affected by the choice of both entrance and exit channel optical parameters. For instance, at 30 MeV bombarding energy, we fixed J_ = 11 h in order to fit the data; we performed another calculation using the parameters given by Hanson et al. ‘) and found the o(J) expansion peaked at lower J-values than previously; then no J-limitation was necessary to obtain good agreement with the data. Fortunately, at higher incident energies, the choice of optical model parameters is not so crucial anymore; for exam-
C. VOLANT et al.
136
ple, calculations at 45 MeV and 60 MeV with these two sets give the same critical angular momenta; this can be understood because expansion (6) in terms of J is limited up to values lower than Jgr in the “C+ 14N channel for which T, are nearly equal to one in both cases. For the 6Li+ 2oNe channel we used the volume absorption optical model parameters obtained from the ‘jLi + “F elastic scattering analysis [ref. ‘)I. Hanson et al. “) found the cross sections reduced by about 15 % for 52 MeV incident energy and increased by about 25 y0 at 76 MeV bombarding energy by using the surface absorption parameters “) instead of the volume ones. As we discussed previously, the a + 22Na channel is very important for the magnitude of the 6Li+ “Ne cross sections. Calculations performed with the optical model parameters used in ref. “) were compared to cross sections calculated with our set of parameters which were taken from Lucas et al. ’ “) and the results are shown in table 3 for different incident energies; there is no significant change. Since at low incident energies (< 30 MeV) the p+ “Mg channel is important, we present in table 4 a comparison of cross sections obtained in the present work by using for this channel a set of parameters given by Hodgson 12) and a set used by Ford et al. “); the differences are within 10 %.
~du/dQ(25”)>Hr &b/sr) \ 20Ne levels Jr
E* (MeV)
Of 2+ 4+
0.0 1.63 4.25
TABLE3 of some Z”Ne levels calculated with different optical parameters a+22Na channel at different 14N incident energies E, = 20 MeV
E, = 40 MeV
for the
E, = 76 MeV
set a
set b
set a
set b
set a
set b
14.4 5.81 0.027
14.57 5.87 0.027
14.4 66.5 95.0
14.65 67.7 96.6
0.6 2.5 4.46
0.6 2.5 4.47
Set a: Set of parameters shown in table 1. Set b: Set used by Hanson er al. 2), V = 54.4 MeV, R = 4.76 fm, a = 0.53 fm, W = 9.80 MeV, RI = 4.76 fm, aI = 0.53 fm, Rc = 3.92 fm.
,,
TABLE4 (,ub/sr) of some Z”Ne levels calculated with different pSZ5Mg at a 14N incident energy of 20 MeV z ONe levels
0+ 2+ 4+
0.0 1.63 4.25
parameters
for
Set b
Set a
E* (MeV)
optical
-14.4 5.81 0.027
15.8 6.37 0.030
Set a: Set of parameters shown in table 1. Set b: Set used by Ford et al. *), V = 53.3-0.55 EC.,. MeV, R = 3.65 fm, a = 0.65 fm, W = 13.5 MeV, RI = 3.65 fm, a, = 0.47 fm, Rc = 3.71 fm.
CRITICAL
ANGULAR
137
MOMENTA
The HF cross sections of the 12C(‘4N, d)24Mg reaction are not sensitive to the entrance optical model parameters since the a(J) expansion is peaked at J-values where the transmission coefficients are nearly equal to one, whatever the optical parameters used. On the other hand they are not significantly affected by the choice of different sets of optical parameters for the d+24Mg channel. For example, we performed calculations with the set shown in table 1 as well as with those used by Hanson ef al. “) and Klapdor et al. 13); t h e d’ff 1 erences are also within 10 %. 3.4.2. The level density parameters. At low incident energies (< 30 MeV) the p+ 25Mg channel has the most important contribution to the denominator G(J). The good agreement observed for the 12C(r4N, d)24Mg reaction at 20 MeV is a check for the level density parameters used for 25Mg. As mentioned already, the contributions to the 12C(14N, 6Li)20Ne cross sections between 40 and 76 MeV incident energies are localized in an angular momentum range where the denominator G(J) is mainly due to the c(+ “Na channel. The choice of the level density parameters a and Y”is rather critical since the absolute magnitudes of the cross sections are quite sensitive to their values (see tables 5 and 6). Most of our calculations for the 12C(14N, 6Li)20Ne reaction have been performed with the set a = 3.67 and y. = 1.6 fm. This value of a is taken from the work of Facchini et al. [ref. ‘“)I by assuming that the law u/A = constant is valid within the Na isotopes (A is the atomic number). The overall agreement observed for the 12C(r4N, d)24Mg excitation functions (fig. 8) is a good justification for this choice, although our recent analysis of the angular distributions of the 12C( 14N, d)24Mg reaction at 35 MeV is more in favour of the theoretical value ’ “) a = 3.13 (r. = 1.6 fm), which fits the experimental absolute cross sections. The J_ values shown in table 2 are derived from the HF analysis when using a = 3.67. If performing a new calculation with a = 3.13, these J,, values would have to be lowered by about 1 h. We feel that uncertainties in the determination of J,, from TABLE5 Effect of the level density parameter a of the afz2Na channel on the performed at different 14N incident energies for some Z”Ne states O+ (0.0 MeV)
20 30 40 76
3.13 3.67 3.13 3.67 3.13 3.67 3.13 3.67
16.65 14.40 65.0 42.3 25.0 14.4 1.7 0.6
2+ (1.63 MeV)
6.7 5.8 229.0 149.0 117.0 66.5 7.2 2.5
&b/x)
4+ (4.25 MeV)
0.032 0.027 136.0 88.8 169.0 95.0 12.66 4.46
138
C. VOLANT et al. TABLE 6
Effect of the level density parameter ~(4 riS = $raAt) in the a+z2Na channel on the HF &b/sr) performed at different 14N incident energies for some ?ONe states
20 40 76
1.4 1.6 1.4 1.6 1.4 1.6
O+ (0.0 MeV)
2+ (1.63 MeV)
4+ (4.25 MeV)
14.24 14.4 19.44 14.4 1.27 0.6
5.74 5.81 90.0 66.5 5.35 2.5
0.027 0.027 130.0 95.0 9.36 4.46
TABLE 7 of measured and calculated cross sections for some states observed l”B(160, 6Li)20Ne reaction at 46 MeV (IIP(~~A~) = 37.2 MeV)
Comparison 2oNe levels Jr
E* (MeV)
2+ 4+ 2+ 2+ 6+
1.63 4.25 7.42 7.83 8.76
,x, dublsr) 65 70 21 20 40
in the
