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Review of high excitation energy structures in heavy ion collisions: Target excitations and three body processes
Nuclear Physics A482 (1988) 245c-262c North-Holland, Amsterdam
245c
REVIEW OF HIGH EXCITATION ENERGY STRUCTURES IN HEAVY ION COLLISIONS: TARGET EXCITATIONS AND THREE BODY PROCESSES
N. FRASCARIA Institut de Physique Nuel~aire, BP 1, 91406 ORSAY CEDEX (France)
A review of experimental results on high excitation energy structures in heavy ion inelastic scattering is presented. The contribution to the spectra of the pick-up break-up mechanism is discussed in the light of the data obtained with light heavy ion projectiles. Recent results obtained with 40 Ar beams at various energies will show that target excitations contribute strongly to the measured cross section.
1. I N T R O D U C T I O N The excitation of collective modes is expected to play important roles in many aspects of the dynamics of nucleus-nucleus collisions. The most direct evidence for the excitations of such states should come from the study of inelastic scattering. Ten years ago, first evidence for the presence of structures at high excitation energy in the inelastic channel from some heavy ion reactions was reported. However due to the complexity of heavy ion reaction mechanisms it was not straightforward to interpret these structures since they could be due not only to nuclear excitations but also to many body processes such as pick-up break-up reactions. In the last few years a large amount of theoretical and experimental work has allowed us to progress towards a deeper understanding of these inelastic spectra. This paper shall be devoted to a review of the most recent experimental results. In parts 2 and 3 the results obtained at low incident energy and the first results obtained at GANIL at intermediate energy will be reviewed in order to show the presence of physical structures in the high excitation energy region. In the two following parts I will present experiments with "light" heavy ion projectiles in which effects of three body processes can he observed and will be discussed. In the last part, preliminary results obtained at GANIL with the S P E G spectrograph will show that the use of a heavy beam such as
4°Ar allows the observation of structures due to target excitations superimposed on the three body continuum. 0375 9474/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
N. Frascaria/ Review of high excitation energy structures
246c
2. E X P E R I M E N T A L OBSERVATIONS AT LOW I N C I D E N T E N E R G Y An i m p o r t a n t a m o u n t of work has been done at low incident energies from 4 M e V / n to 11 M e V / n 1-0 :the s t u d y of s y m m e t r i c systems followed by the s t u d y of very a s y m m e t ric systems, inclusive e x p e r i m e n t s a n d coincidence experiments. These results are now p u b l i s h e d 1.2,5 a n d I would like j u s t to review the m a i n characteristics of the structures• At these low incident energies : • the s t r u c t u r e s are observed in the inelastic and few nucleon transfer channels for angles near or slightly before the grazing angle. • their positions are i n d e p e n d e n t of angle. • their w i d t h increases with excitation energy. • the s t r u c t u r e s at low excitation energy (E, _< 40 MeV) are peaked at the grazing while the ones at E= > 40 MeV are more forward peaked. In the 4°Ca +40 Ca reaction studied at 4, 7 and 10 M e V / n , the positions of the b u m p s are i n d e p e n d e n t of incident energy in the energetically allowed excitation energy range. All these e x p e r i m e n t a l features suggest t h a t the s t r u c t u r e s corresponding to well defined excitation energies are excited mainly t h r o u g h a direct process a n d t h a t we are dealing w i t h an excitation of the target nucleus. However the d a t a concerning these s t r u c t u r e s were restricted to a limited n u m b e r of systems a n d b o m b a r d i n g energies a n d more systematic studies were necessary to resolve several questions concerning t h e i r nature.
3.
F I R S T E X P E R I M E N T A L RESULTS W I T H AN Ar B E A M AT I N T E R M E D I A T E
ENERGY F r o m D W B A calculations giant resonance cross sections from heavy ion inelastic scattering are expected to be considerably en-
z
4°Ar
9°Zr
8
1000 --3°<8Lob <2.7° ~,~'
2'
1500
/~'
!r :
into t h e i r excitation m e c h a n i s m is to perform e x p e r i m e n t s at higher incident energies. T h e
z
~,
Ein c = L4 MeV/n /,0 50MeV Ar 67 58 I
isoo
h a n c e d at high incident energy• If the high energy s t r u c t u r e s can be related to collective excitations one way to get a deeper insight
+
1250
I"
¢111:
,J,*'*~W,, ,÷~
:
: 1500
5oo-
giant resonances are known to be a general
#~,~*
~ 2.8°<8Lob <32 °
,,*"
¢,.
p r o p e r t y of nuclei• T h u s a b e t t e r knowledge of t h e evolution of these s t r u c t u r e s with mass n u m b e r is required• Therefore we have per-
0:
750
300-~ , ~
,~o~
3.8o
formed e x p e r i m e n t s at GANIL at intermediate energy.
E x (MeV) 70
50
30
Fig. 1. Inelastic 40At + 90Zr spectra obtained at 3 different angles presented in a linear scale with 1 MeV/channel-binning.
N. Frascaria/ Review of high excitation energy structures
247c
A first step was the study of the 4°Ar +40 Ca, 9°Zr, 12°Sn and 2°SPb reactions at 44 M e V / n using a time of flight spectrometer built for these experiments ~. Data were taken for angles close to the grazing angle in each reaction. In fig.(1) are displayed three inelastic 4°Ar +90 Z r spectra obtained at different angles around the grazing angle (Ogr = 2.9°). At angles close to and slightly before the grazing the bumps show up clearly in the spectra up to 60 MeV and they are best observed at forward angles. A very rapid decrease with angle is observed for the structures and for large angles (~ 4 °) they have completely disappeared. These structures have a small cross section and are superimposed on a large physical background. In order to obtain more quantitative information on the high energy structures statistical analyses such as auto-correlation and cross-correlation analyses were carried out on the inelastic energy spectra of the 4°Ar +90 Z r reaction following the methods developed in ref. 8. In these calculations, the structures are analyzed as fluctuations around an average cross section which is defined by the sliding average method. 1
~+~
< a(Ei) >= -2A- + 1 j=i--z~o ( z i ) where A corresponds to an averaging interval of :E 6 MeV in the present case. The auto correlation function is then defined by
C(~)
(a(E + c ) - < a(E + ~) > ~ca(S)-, < a(S) >_~j> < a(E + e) > < a(E) >
In fig. (2) is displayed the result of the auto-correlation calculations for different energy spectra over the excitation energy range 36 < E* < 82 MeV. In such a correlation function the first channel corresponding to the correlation between two unshifted spectra indicates the strength of statistical fluctuations. For small angles the first shoulder observed in the correlation functions demonstrates the presence of structures in the spectra. The strong b u m p centered around e = 9 MeV shows that the experimental structures are regularly spaced by about 9 MeV. Even if the correlation becomes rather weak the presence of the following bump centered around ~ = 20 MeV indicates that at least three structures are present in the studied excitation energy range of the original spectra. At angles larger than about 3.80 where the structures are not seen in the experimental spectra the correlation function exhibits (see fig. (2.1)) a very different pattern. None of the above features is observed. A cross correlation analysis has been carried out following the same method in order to obtain the degree of correlation of the structures observed in the spectra taken at different angles (see fig.(2.2)). The cross correlation analysis was done by shifting the first s p e c t r u m with respect to the second one in order to find the different cross correlation maxima. These maxima are related to different bumps in the spectra and their position gives the average spacing between the bumps. In fig. (2.2) three cross correlation maxima are clearly observed in all cross correlation curves. Again the average spacing between
N. Frascaria / Review o f high excitation energy structures
248c
4°Ar+9OZr E inc =/,4 MeV/n
u \1
o)
Ax
4°Ar
2.3°< 8Lob < 2.8o 0.5 0
U 0.1
U1] n A
n. ~ ] ~ n
hill.
40Ar+90Zr ~ 4 0 A r
.Era.
2.o.4
eLob: 2.7~nd 8Lob =2.~
(-~
0.2 (o)
!
2J~
0
-0.2 -0.5
~0.1
1
b)
3.8°< BLab
eLob:2.8*oncl 8to b :3.2°
-0.40,4~
0.5 0.2 0 --0.2
0.20 -&2
~] 0.5Jl
Honte CarLo
0.1
c)
IU-u.,,"u-u,-', u-u,., ll Uk
0
u
-0.4.
0.2 o
(c]
MonteCorto
JlYJl
-0.1
0
10
Fig. 2.1
20 ~(MeV)30
0,2 o -20
-10
0
10 E.MeV.20( )
Fig. 2.2
Fig. 2.1. Auto-correlation functions for two 40At + 90Zr inelastic spectra and for a random fluctuation spectrum (see ref. 7}. Fig. 2.2. Cross-correlation functions for 40At + 90Zr inelastic spectra a) between eLa b = 2.7 ° and eLa b = 2.8 ° b) between 9La b = 2.8 ° and eLa b = 3.2 ° c) between two random fluctuation spectra (see ref. 7).
N. Frascaria/ Review of high excitation energy structures
249c
these m a x i m a is around 10 MeV which is in agreement with the auto-correlation analysis. Moreover the fact that the cross correlation functions exhibit a m a x i m u m for e = 0 indicates that the structures lie at the same excitation energies, at all angles where they are observed. In conclusion, these analyses quantitatively confirm the existence of physical structures regularly spaced in the excitation energy range of 36 to 82 MeV. As mentioned before, to understand the mechanism of excitation of these physical structures the knowledge of the evolution of the structures with incident energy and with target mass is primordial. A first attempt consisted in the study of the influence of the target. The inelastic scattering of 44 M e V / n 4°Ar from 4°Ca, 9°Zr, l~°Sn and 2°sPb targets has been investigated and the resulting spectra are displayed in fig.(3). Structures are observed in the high excitation energy region in all inelastic channels. These structures are not found at the same excitation energies in the four spectra. The structures seem to be a general property of nuclei and clearly evolve as a function of the target mass. This point will be developed later.
Z'0Ar +90Zr Z
~,
LN.
O O
I00(
Z O O
50C 200 100
60(
400
200 750 500
60( 25(3 108 0
Ex(MeV)80
70
60
50
/~0
30
20
Fig. 3. Inelastic 40Ar spectra from 208pb, 90Zr and 40Ca presented in linear scale with 1MeV/channel binning. (For more details see ref. 7)
Ex (MeV)
70
50
30
Fig. 4. Inelastic 40Ar + 90Zr spectra measured a) at 44 MeV/n b) at 33 MeV/n. Spectrum c) illustrates the effects of the different backgrounds (see ref. 7).
2 50c
N. Frascaria / Review of high excitation energy structures
In these experiments performed with the time of flight system, the influence of the incident energy has also been investigated by studying the 4°Ar +90 Z r reaction at 33 MeV/n.
The comparison of the inelastic channels obtained in the A r ÷ Z r reactions
at 44 MeV and 33 M e V / n is given in fig. (4). At least the bumps at 43, 50 and 58 MeV (+2 MeV) are clearly observed in both spectra while the bump at 66 MeV is less visible in the 33 M e V / n spectrum. This can be an effect due to the rapid fall off of the background in that region (see ref. 7). Thus, for the bumps up to 58 MeV this result is a strong element in favour of a target excitation. But it is well known that three body processes can contribute strongly to the cross section of the studied spectra and consequently before drawing any conclusion a careful study of these processes is necessary. 4. P I C K - U P B R E A K - U P P R O C E S S In this process the projectile picks up one or more nucleons from the target and subsequently decays in flight by the emission of one or several light particles. For example the inelastic spectrum of the 4°Ar+9°Zr reaction will be contaminated by the following two step processes : 40Ar ÷90 Z r
--~ 41Ar, +89 Z r ] ) 4°Ar + n -~ 4aK, +89 y
4OAr +9o Z r
! - -
~ 4°Ar + p
etc...
It is very instructive to calculate 9 by simple kinematical considerations the apparent excitation energy and the kinematical broadening of the bump generated by such a process and to study its dependence on incident energy and on target and projectile masses. This calculation can be done by comparing the two following reactions :
and
1+2 1+2
-~ -~
3~+4 ' 3+4 ]. ) 3 ~÷ 5
As in heavy ion reactions M5 << M4 and M5 << M3 the apparent excitation energy and the width of the pick-up break-up component contributing to the inelastic spectrum can be written : E;p~
MSE
N tab+S.~(4')+ < E 5 > + E r
F Y
F-~4~/M5
E t?b
M~
N. Frascaria/ Review of high excitation energy structures
251 c
where < E5 > is the average kinetic energy of the emitted light particle in the frame of the projectile-like fragment, $5(4~) is the separation energy of the light particle in the target and E~ is the excitation energy of the target-like nucleus. From these formulae we can deduce that E~:p depends very little on the nature of the target. Thus for a given projectile at a given incident energy, the pick-up break-up contribution is nearly the same for reactions on different targets. The most important term in these formulae is the one proportional to E~b/M3 which implies that E~p and I' depend strongly on the beam energy per nucleon. Thus a variation of the bombarding energy can provide an unambiguous discrimination between target excitation and decay products of excited projectile fragments. This is an important point for the following. In order to describe in detail the energy 40Ar+90Zr spectra of the various transfer evaporation 8 Ein c :/J. MeV/n channels a Monte Carlo calculation has been ~000, 2.3*< 8Lo b < 2.7° performed 9 for the studied reactions using /.0Ar / the code LILITA 10 which assumes a purely y statistical decay process in the framework of the Hauser Feschbach theory where the light particles are emitted isotropically in the cm of the fully accelerated fragment. As
~00 '
-
~
~
the first step of the reaction is not known, this calculation necessitates several assumptions which are discussed in detail in ref 9. Fig. (5) displays the inelastic 4°Ar +~0 Zr spectrum compared to the result from the
~0
LILITA calculation. In this calculation all the evaporation channels have been taken
Fig. 5. Comparison of the inelastic 40At + 90Zr spectrum with the pick-up break-up calculation described in text.
1600
1700ELob(HEY)
into account. The resulting curve is normalized to the experimental spectrum in order to indicate the maximum contribution consistent with the data. From this comparison, one can conclude that : i) the widths of the calculated contributions are much larger than the ones of the observed structures and thus this process contributes only to the background of the spectra ii) the evaporation channels of more than one nucleon contribute to a region where the experimental cross section is very small and where no structures are observed. Before going into the details of the most recent results obtained at GANIL, let me present and discuss results from other laboratories using "light" heavy ion beams. 5. EXPERIMENTAL RESULTS WITH "LIGttT" HEAVY ION PROJECTILES Recently several studies of inelastic scattering of light heavy ions at intermediate
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N. Frascaria / Review of high excitation energy structures
energy have been performed. Two of them, the studies of the 2°Ne +208 Pb reaction 1, at 30 M e V / n and of lrO +2o8 Pb at 22 M e V / n 12 concluded to the absence of physical structures in the high excitation energy region. However these conclusions must be taken with caution since in both experiments several structures were observed but attributed to defects in the detection systems and moreover a very recent study 13 of 2°Ne inelastic scattering on 2°Spb has demonstrated that the conclusions of ref. 11 were incorrect and that physical bumps are present in the ZONe ~_~0sPb spectra at 30 MeV/n. Let me now present and discuss the two recent experiments 160 +20s Pb performed at 22 and 25 M e V / n (Oak Ridge) 14 and ZONe +2os pb,9OZr at 25 and 30 M e V / n (MSU) 13 5.1. The 160 + 2°SPb experiment 14
60oo
This reaction was studied at two slightly different incident energies. In fig. 6 are displayed the two inelastic channels from this reaction at 22 and 25 MeV/n. Together
~o0
with the G.Q.R. and two peaks at 13.6 and 17 MeV two bumps are observed at approximately 25 and 40 MeV excitation energies. These two bumps have a cross section only
4o0o
a factor ~ 2 smaller than the G Q R and be-
looo
LILITA as described in paragraph 4 cannot reproduced the two observed bumps. This point will be discussed in section 5.3. 5.2. The 2°Ne + ~°Spb,9°Zr experiment 13
J
~
=
L
~ ~.
~t ill
°
energies are shifted by ~ 2.5 MeV. Clearly these structures cannot be identified with
up break-up component calculated with
,
V
:re0 - 4o0 M,V-2.SI~V~ moo
kinematical evolution predicted for a pickup break-up reaction. Obviously the pick-
,
~ 3o00
tween the two experiments their excitation
high excitation energy states of the target nucleus since the observed shift follows tile
,
~'°spbd60, 160')208pb
0-,2°
l
~ .
.
.
I
t
.
E16 C = 350 MeV
....
~ I
I
I
Oi. b " 14 ° i i I
oss so 4s 4o 3s a0 2s EXCITATION
I
/f
y .
~,~ ~ I
11t
~ -..~J
~ .
~-
ENERGY
20 is
lo
(MeV)
Fig. 6. Inelastic spectra from 160 + 208pb reaction from ref. 14. The solid line represents the calculation of the double humped pick-up break-up component arbitrarily normalized to the data (see text).
The inelastic scattering of 2°Ne on 9°Zr and 2°8Pb has been studied at 500 and 600 MeV incident energies. High statistics spectra were measured at the grazing angle using a magnetic spectrograph. In fig. (7) are displayed the inelastic spectra obtained in these reactions. In the case of the Pb target, structures are clearly observed in the inelastic spectra at the two incident energies but they appear at different excitation energies. A cross correlation analysis of these data was carried out for both studied systems.
In
the case of the Ne + Pb system, the cross correlation function is strongly negative for unshifted spectra, indicating that the structures observed at 600 MeV and 500 MeV incident energies are out of phase (see fig. 8). It takes its m a x i m u m when one of the spectra is shifted by about 5 MeV relative to the other. Again this corresponds exactly
N. Frascarial Review of high excitation energy structures
ZONe + 209pb
253c
20Ne+9°Zr
,x,,
......
)
,=,
/.0000
+,
~
g
+1
g
+,
a)
g 3o0oo ',
>= .i
g "-. m
30000
20000
u o
"" ' Co
IOOOO
E Z
5000
30
40 50 Ex (MeV)
60
30
70
40
50
60
70 Ex {MeV)
Fig. 7. Inelastic scattering spectra at the grazing angle from the reactions 20Ne + 208pb and 20Ne + 90Zr at 500 and 600 MeV incident energies (for more details see ref. 13).
®2.o Ne 6 0 0 M e V
0N e 600MeV 2ONe 5 0 0 M e V
c,,I
2 ° N e 500MeV ( J' -
1
c~
o -I
, ,,
1
0 ,l
,,,,
-4-2024 A (MeV)
-1
I
I
f
,
i
-4-2
i
i
i
i
I
0 2 L, A
(MeV)
Fig. 8. Cross-correlation functions between the spectra taken at 25 MeV/n and 30 MeV/n for the 20Ne + 208pb (a) and 20Ne + 90Zr (b) reactions.
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N. Frascaria / Review of high excitation energy structures
to what is expected from a quasi projectile excitation. A different result is obtained for the ZONe +90 Zr inelastic spectra. The correlation function is rather flat and close to zero for unshifted spectra (see fig. 8). Let me now discuss these results in terms of pick-up break-up effects, 5.3. Discussion In the pick-up break-up calculations discussed in the preceding paragraph, continuous excitation energy and spin distributions were postulated for the levels excited just above the emission threshold of the quasi projectile. If now we assume that due to momentum matching conditions a level just above the particle emission threshold is highly excited in the quasi-projectile giving rise to a strong angular momentum alignment the light particles will be emitted following a Legendre Polynomial Pe,~ (cos 0) distribution. A complete calculation 9 shows that the transfer evaporation contribution is then split into two peaks separated by the width F. This double humped contribution follows the same evolution with incident energy and target mass as described before for the flat pick-up break-up component. 3+ L =- 2 level at 5.08 In the case of the 160 +208 Pb reaction, if we assume that the ~ MeV is strongly excited in 1tO with an aligned angular momentum one can reproduce completely the data 9 (fig. (6)).
As already observed in the ~60 +z0s Pb experiment
the shift between the two main bumps observed at 25 MeV/n and 30 MeV/n in the 2°Ne -I-2o8 Pb inelastic channel roughly corresponds to the difference between the two incident energies per nucleon. A complete pick-up break-up calculation shows that these peaks could correspond to the decay of an unbound state in 21Ne which emits a neutron of 0.4 MeV (see ref. 13). One can notice that this phenomenon occurs for quasi projectiles such as 170 and 2lNe having a low level density above the emission threshold. In the case of heavy quasi projectiles such as 41At discussed before, the neutron threshold is high and the level density above this threshold is very large. Consequently in that case the probability to excite preferentially discrete states in the region above the emission threshold is certainly small. But it is not straightforward to appraise the probability of the presence of this double humped structure. It has been shown that, whether double humped structures are present or not, the centroid and the width of the pick-up break-up contribution depend strongly on the beam energy per nucleon. Thus, a clean method to conclude between target excitation and pick-up break-up reactions is to study a given system at two different incident energies.
Even in this case the change of matching conditions
between two very different incident energies could modify the relative population of tile quasi projectile states. Consequently a definite conclusion concerning such a mechanism requires a comparison of two experiments where the bombarding energy per nucleon is varied by approximately half of the mean interval between two observed bumps. 6. S.P.E.G EXPERIMENTS In this paragraph, let me present very recent and preliminary data we have obtained
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N. Frascaria / Review o f high excitation energy structures
at GANIL using the magnetic spectrograph SPEG. The aim of this experiment was to compare with high resolution and high statistics the inelastic scattering of 4°Ar on 9°Zr at 41 M e V / n and 44 M e V / n in order to distinguish between excitations of the projectile and the target fragments. The m o m e n t u m range of the spectrograph is sufficiently large so that the entire excitation energy range of interest (~ 100 MeV) could be covered at one magnetic field setting. The total aperture of the spectrograph is 4 ° . The emitted fragments were detected by the standard focal plane detection system consisting of two drift chambers for position measurements and two rnulti-anode ionisation chambers for particle identification.
A time of flight measurement was taken between a large area
parallel plate counter and the cyclotron R.F, An unambiguous mass, atomic and nuclear charge identification of the fragments in the vicinity of 4°Ar was obtained. The first remarkable result of this experiment is the large peak to continuum ratio
GQR i
40Ar + 90Zr
Z.0Ar
"~ "~o 5000-
59 50 66 t ~
4500"
40E I j : : : : M
O
eV/n
4% :
5000-
o
/
LEO~IJ 3000-
~000
¢t/'lt GQR
i
7500
2500-
I~
-5000
]
-2500
2000
20
-8000
ex =2
3000-
2%
8o (..9 12000
~1 ~" ~'
j ~;~#tl~t~#~'
Einc :41Mev/n
Ex (MeV)
..i-,
Ex=36MeV 1/
Einc : ''MeV/n
io
Fig. 9. Low excitation energy region (0 < Ex < 20 MeV) of the inelastically scattered 40At spectrum from 90Zr taken at the grazing angle.
>~
90 70 Ex(MeV)
50
30
Fig. 10. Inelastic 40Ar + 90Zr spectra measured at 41 MeV/n and 44 MeV/n with SPEG with two different representations (see text). The arrows indicate the positions of the bumps observed in the time of flight measurement at 44 MeV/n for the same system.
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N. Frascaria / Review o f high excitation energy structures
observed for the giant quadrupole resonance which is considerably e n h a n c e d at t h e studied energy c o m p a r e d to heavy ion e x p e r i m e n t s at lower incident energy (see fig. 9). In fig. (10) are displayed the two inelastic spectra at 44 M e V / n a n d 41 M e V / n incident energies, b o t h w i t h two different energy binnings : 250"keV/channel a n d 500 k e V / c h a n n e l in order to observe more clearly the b r o a d s t r u c t u r e s at high excitation energy. At first glance the s p e c t r a at b o t h incident energies are very similar a n d exhibit s t r u c t u r e s up to ~ 60 M e V excitation energy. In order to insure t h a t these small cross section fluctuations are not due to differential n o n linearities of the position sensitive drift c h a m b e r s m e a s u r e m e n t s were performed with t h r e e different settings of the magnetic field of the s p e c t r o g r a p h at b o t h incident energies. To check t h e consistency of the excitation energy s p e c t r a at different (Bp) cross CROSS CORRELATION
1llIj11111111 CROSS CORRELATION
1.0
^r.Zr 41MeVlu
0.5
0.5
Tlltttr1tttt tt ttttttt
0,0
0.0
tt.o..o
"0.5
t "20
2.BOo3.Bp -10
0 ¢ (MeV}
10
-0.5
20
I
I
-20
I
I
I
I
I
I
-I0
I
I
I
I
I
I
L
I
0 ¢ (MeV)
I
I
I
I
I
~
I
I0
20
Fig. 11. Cross-correlation functions between spectra taken at two different settings of the magnetic field: on the left at 44 MeV/n, on the right at 41 MeV/n. The errors bars are calculated following ref. 8.
correlation analyses were carried out a n d are displayed in fig. (11). T h e fact t h a t the cross correlation functions show a m a x i m u m for eel < ] MeV clearly d e m o n s t r a t e s t h a t t h e m a i n oscillations observed in the s p e c t r a have a physical origin. T h e u n c e r t a i n t y of t h e energy c a l i b r a t i o n is estimated to be less t h a n 1 MeV. Now, we shall c o m p a r e the two inelastic spectra o b t a i n e d at 44 M e V / n a n d 41 M e V / n (fig.
(12)).
These s p e c t r a are preliminary a n d the full statistics have not yet been
analysed. Nevertheless the result is clear : the b u m p s are not shifted a n d show up at t h e same position in b o t h spectra. In the u p p e r p a r t of fig. (12), these two s p e c t r a are s u m m e d for a same n u m b e r of counts a n d in the resulting s p e c t r u m the three observed b u m p s remain. Moreover a cross correlation analysis between the two s p e c t r a taken at 44 M e V / n a n d 41 M e V / n has been carried out. As one can see in fig.
(13)
a strong
m a x i m u m is observed for the unshifted s p e c t r a (~ -- 0) showing t h a t the s t r u c t u r e s are in p h a s e at b o t h incident energies. A s u p p l e m e n t a r y test to show t h e accuracy of this result is given by the c o m p a r i s o n
N. Frascaria / Review of high excitation energy structures
CORRELATION
CROSS 1
>~
>~
•-
.T.
O3 I..-
O3 I--
Z
D O
(.9
1
1
1
1
,
,
,
,
,
l
l
T
J
,
l
J
Ar • Zr 0.5
J
,
44
,
,
,
-
-
MeVIue 41MeVlv
llllllllllIlilllll
z
O
1
257c
0.0
(D -0.5
I I L I I l . L l o l i i i l l l l l l l l i0l ~ o I . 2 0 lO ¢ {HeV}
10 000
7 500
800C
5 000
Fig. 13. Cross-correlation function between the two spectra of the lower part of fig. 12.
Z,OAr +90Zr
o
I.500
450C
>~
t
4OAr
~
o u
6"/50"
#
7°Ex (MeV)5°
4500-
/
/ /z 2250
/
:
.
-5750
(o,/ ~e ' ~ ' J / ,.'
2250-
Fig. 12. Lower part: comparison between the inelastic spectra from the 40Ar + 90Zr reaction measured at 44 MeV/n and 41 MeV/n. Upper part: sum of a 41 MeV/n and 44 MeV/n spectrum having both approximately the same number o f counts.
-9000
/
.4500
(b)/'
./
/
Ex(MeVlO
2250
'/'o
5'0
3'o
Fig. 14. a) target excitation case: same as upper part of fig. 12 with a different energy binning b} quasi-projectile case (see text}.
258c
N. Frascaria/ Review of high excitation energy structures
of the two spectra displayed in fig. 14. Spectrum a) is obtained by summing the spectra at 44 and 41 MeV/nucleon for a same number of counts. The bumps observed in these two spectra are clearly visible in the sum spectrum a) which is expected if they are due to target excitation. On the other hand structures produced by pick-up break-up reactions are expected to shift by 3 MeV between 44 and 41 M e V / n (see paragraph 4). To investigate this last hypothesis, spectrum b) presents the sum of the 44 M e V / n data and the 41 M e V / n spectrum shifted upwards in excitation energy by 3 MeV. Within the statistical error bars spectrum b) is flat showing that the contribution to the structures of double humped pick-up break-up components, if it exists at all, is very weak. Moreover the structureless spectrum obtained in this case shows that the 3 M e V / n energy shift between the two incident energies provides a meaningful test to conclude between target excitations and three body reactions. In fig. (15) are displayed the four inelastic spectra obtained in the A r + Z r reaction
Z,0Ar + 90 Z r
415MeV÷ ÷ + (' Ex:59' ** l ,
4OAr
%
o (J
51, !
l. ,j ;
10000-
:C:in~=4iMelv/n
2co~ (_)
~'
•3600
800( •
#~+!
SPEG:
+
+
.+~, !Einc=4&MeV/n +
600C
2400
.?#
,
)*#,+ )
.#WWTF(x~-) I~ "
r ei~c=41NeV/n .
,~'#I
4500
: Ii,
+
~
,
t
'
J i!
,÷
.~# @'"+, 1200
; I! '
Eihc=33MeV/n
"
#
J
E~'IMe'V) 7'o
'
5'0
'
3'0
'
Fig. 15. Comparison between the inelastic spectra from the 40At + 90Zr reaction measured at 33 MeV/n {time of flight measurement}, 41 MeV/n (SPEG measurement• ) and 44 MeV/n using the two different set ups.
259c
N. Frascaria / Review o f high excitation energy structures
at 33 M e V / n , 41 M e V / n and 44 M e V / n with two different set ups - the time of flight (TF) discussed previously and SPEG. Clearly the bumps show up at the same position at the three different incident energies. One can notice that the two spectra obtained with T F and S P E G are identical and differ only by the number of counts which is a factor 6 larger in the S P E G experiment. 7. D I S C U S S I O N If we recapitulate now all the data obtained with the 4°Ar beam at GANIL it has been shown that i) in the case of the Z r target the positions of all the structures are confirmed with high statistics in the S P E G experiment. This gives confidence in the time of flight measurements presented before ii) the results obtained at 33 M e V / n , 41 M e V / n and 44 M e V / n for the A r + Z r reaction point to an interpretation of the structures in terms of target excitation iii) a clear evolution of the structures with target mass is observed. Thus, from the time of flight data we have tried to extract the different positions of these bumps in the four studied systems. Different methods have been used (see ref. 7) consisting in different background subtractions and a double Fourier transform analysis. A compilation of these results taking into account all the measured angles is shown in fig. (16). The error bars reflect the dispersion on the excitation energies given by the different
A-~'~
/,'LE.O.R /
,,,,o.oi / G.OR
104 (H,ED.R)
/
128 156 12phone //(L=5}
186 (3phonons)
~ /
~(4~onons)
/
-e--~
225 _(L=/,)
~q,
302
~ ~ ~ ~ ( S p h
o 2L
Sn
~
~
..... )
-*/- .+/--.-/~J 1.7~ ~+j
6'0
4 Ex(M~V)~
Fig. 16. Excitation energies of the structures observed in the inelastic channels of the four reactions (40At + 40Ca, 90Zr, 120Sn, 208pb) plotted as a function of target mass A- 1 / 3 Tentative interpretations of the different states are given. analyses and the uncertainties due to statistical fluctuations and energy calibrations. An a t t e m p t was made to reproduce the positions of the structures with different laws of the type E~c~A P but the only consistent representation of the data is given by A -1/3 laws as
260c
N. Frascaria/ Review of high excitation energy structures
presented in fig. (16). In such a representation, the points can be joined with surprising consistency by straight lines passing through 4 data points and the origin. The slopes of these lines in units of M e V / A -1/3 are also indicated. The second slope corresponds to the well known G.Q.R. The first and the third ones can be tentatively associated with the low excitation and the high excitation octupole resonances (LEOR and HEOR). The others are not known and we can try to conjecture what might be the nature of these excitations. To understand the origin of these bumps a large amount of theoretical work has been carried out during these last years 7. Up to now the most consistent representation of the data is given by the multiphonon calculation 6,15 where one supposes the excitation of multiphonon states built with the giant quadrupole resonance. This implies that the multiphonon energies are multiples of the G.Q.R. energy. As one can see in fig. (16) the 128, 186 and 261 M e V / A -1/3 slopes are in surprisingly good agreement with such a picture corresponding respectively to what is predicted for 2, 3 and 4 phonons but all the data cannot be completely interpreted by this model and the nature of these structures remains an open question. 8. CONCLUSION From the presented work, we have learned that projectile effects can be very important in some heavy ion collisions using "light" heavy ion projectiles leading to a highly excited double humped structure in the inelastic spectra. Consequently, if small target excitations are present in the spectrum this effect will not allow to observe them. With a heavy projectile such as 4°Ar, it has been shown that structures are observed in all studied target nuclei and these data are consistent with an excitation of the target. Light particle-heavy fragment coincidence experiments should now provide a unique tool to disentangle the contributions of projectile and target excitations in heavy ion inelastic spectra and will allow a further step towards the understanding of these complex reaction mechanisms.
Acknowledgements : I wish to acknowledge my colleagues D. Beaumel, Y. Blumenfeld, Ph. Chomaz, J.P. Garron, J.C. Roynette, T. Suomij£rvi, J. Barrette, B. Berthier, B. Fernandez, J. Gastebois and W. Mittig for many fruitful discussions and for allowing me to present our unpublished results. Special thanks are due to Y. Blumenfeld for a careful reading of the manuscript. I acknowledge D. Grialou for the elaboration of the manuscript.
References 1) N. Frascaria, C. Stephan, P. Colombani, J.P. Garron, J.C. Jacmart, M. Riou, L.
N. Frascaria / Review o f high excitation energy structures
261 c
Tassan-Got, Phys. Rev. Lett. 39(1977)918 2) N. Frascaria, P. Colombani, A. Gamp, J.P. Garron, M. Riou, J.C. Roynette, C. St6phan, A. Ameaume, C. Bizard, M. Louvel, Z. Phys. A294 (1980) 167 N. Frascaria, Proceedings of the Int. Conf. on Nucl. Phys. (Trieste}, Nucl. Phys. C.H. Dasso, R.A. Broglia, A. Winther EDS North Holland, Amsterdam (1982} 617 3) A.C. Mignerey, K.L. Wolf, M. Breuer, B.G. Glagola, V.E. Viola, J.B. Birkelund, D. Hilscher, J.R. Huizenga, W.V. Shroeder, W.W. Wilcke, Proc. of the Int. Conf. on Nucl. Phys., Berkeley, California (1980) 4) S. Pontoppidan, P.R. Christensen, D. Hansen, F. Videback, H.C. Britt, B.H. Erkkiza, Y. Patin, R.H. Stokes, M.P. Webb, R.L. Ferguson, E. Plasil, G.R. Young, Phys. Rev. C 28 (1983) 2299 5) Ph. Chomaz, N. Fra~caria, Y. Blumenfeld, J.P. Garron, J.C. Jacmart, J.C. Roynette, W. Bohne, A. Gamp, W. von Oertzen, M. Buenerd, D. Lebrun and Ph. Martin, Z. Phys. A318 (1984) 41 6) Ph. Chomaz, Y. Blumenfeld, N. Frascaria, J.P. Garron, J.C. Jacmart, J.C. Roynette, W. Bohne, A. Gamp, W. von Oertzen, N.V. Giai and D. Vautherin Z. Phys. A 319 (1984) 167 Ph. Chomaz, th~se de 3~me cycle, IPNO Th 84-01 7) N. Frascaria, Y. Blumenfeld, Ph. Chomaz, J.P. Garron, J.C. Jacmart, J.C. Roynette, T. Suomij~rvi and W. Mittig, Nucl. Phys. A474 (1987) 253 and references therein 8) P.J. Dallimore and I. Hall, Nucl. Phys. 88 (1966) 193 A. Richter, Nuclear Spectroscopy and reactions, Academic Press New York and London, 1974 M.G. Kendall and A. Stuart, Advanced theory of statistics, vol. 3, London (1976) 3rd ed. 9) Y. Blumenfeld, J.C. Roynette, Ph. Chomaz, N. Frascaria, J.P. Garron and J.C. Jacmart, Nucl. Phys. A 445 (1985) 151 10) J. Gomez del Campo and R. Stokstad, Oak Ridge, National Laboratory, report ORNL TM 7295 11) M. Buenerd, J. Chauvin, G. Duhamel, J.Y. Hostachy, D. Lebrun, P. Martin, P.O. Pellegrin, G. Perrin and P. de Saintignon, Phys. Lett. 167B (1986) 379 12) F.E. Bertrand, R.O. Sayer, R.L. Auble, M. Beckerman, J.L. Blankenship, B.L. Burks, M.A.G. Fernandes, C.W. Clover, E.E. Gross, D.J. Horen, J. Gomez del Campo, D. Shapira and H.P. Morsch, Phys. Rev. C35 (1987) 111
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N. Frascaria / Review o f high excitation energy structures
13) S. Fortier, S. Gal~s, S.M. Austin, W. Benenson, G.W. Crawley, C. Djalali, J.H. Lee, J. Van der Plicht and J.S. Winfield, Report IPNO DRE 87-22, Orsay (1987) and to be published in Phys. Rev. C 14) T.P. Sjoreen, F.E. Bertrand, R.L. Auble, E.E. Gross, D.J. Horen, D. Shapira and D.B. Wright, Phys. Rev. C29 (1984) 1370 15) Ph. Chomaz, N.V. Giai and D. Vautherin, Preprint IPNO Th 86-32 Orsay (1986), Nucl. Phys. (to be published) and references therein.
245c
REVIEW OF HIGH EXCITATION ENERGY STRUCTURES IN HEAVY ION COLLISIONS: TARGET EXCITATIONS AND THREE BODY PROCESSES
N. FRASCARIA Institut de Physique Nuel~aire, BP 1, 91406 ORSAY CEDEX (France)
A review of experimental results on high excitation energy structures in heavy ion inelastic scattering is presented. The contribution to the spectra of the pick-up break-up mechanism is discussed in the light of the data obtained with light heavy ion projectiles. Recent results obtained with 40 Ar beams at various energies will show that target excitations contribute strongly to the measured cross section.
1. I N T R O D U C T I O N The excitation of collective modes is expected to play important roles in many aspects of the dynamics of nucleus-nucleus collisions. The most direct evidence for the excitations of such states should come from the study of inelastic scattering. Ten years ago, first evidence for the presence of structures at high excitation energy in the inelastic channel from some heavy ion reactions was reported. However due to the complexity of heavy ion reaction mechanisms it was not straightforward to interpret these structures since they could be due not only to nuclear excitations but also to many body processes such as pick-up break-up reactions. In the last few years a large amount of theoretical and experimental work has allowed us to progress towards a deeper understanding of these inelastic spectra. This paper shall be devoted to a review of the most recent experimental results. In parts 2 and 3 the results obtained at low incident energy and the first results obtained at GANIL at intermediate energy will be reviewed in order to show the presence of physical structures in the high excitation energy region. In the two following parts I will present experiments with "light" heavy ion projectiles in which effects of three body processes can he observed and will be discussed. In the last part, preliminary results obtained at GANIL with the S P E G spectrograph will show that the use of a heavy beam such as
4°Ar allows the observation of structures due to target excitations superimposed on the three body continuum. 0375 9474/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
N. Frascaria/ Review of high excitation energy structures
246c
2. E X P E R I M E N T A L OBSERVATIONS AT LOW I N C I D E N T E N E R G Y An i m p o r t a n t a m o u n t of work has been done at low incident energies from 4 M e V / n to 11 M e V / n 1-0 :the s t u d y of s y m m e t r i c systems followed by the s t u d y of very a s y m m e t ric systems, inclusive e x p e r i m e n t s a n d coincidence experiments. These results are now p u b l i s h e d 1.2,5 a n d I would like j u s t to review the m a i n characteristics of the structures• At these low incident energies : • the s t r u c t u r e s are observed in the inelastic and few nucleon transfer channels for angles near or slightly before the grazing angle. • their positions are i n d e p e n d e n t of angle. • their w i d t h increases with excitation energy. • the s t r u c t u r e s at low excitation energy (E, _< 40 MeV) are peaked at the grazing while the ones at E= > 40 MeV are more forward peaked. In the 4°Ca +40 Ca reaction studied at 4, 7 and 10 M e V / n , the positions of the b u m p s are i n d e p e n d e n t of incident energy in the energetically allowed excitation energy range. All these e x p e r i m e n t a l features suggest t h a t the s t r u c t u r e s corresponding to well defined excitation energies are excited mainly t h r o u g h a direct process a n d t h a t we are dealing w i t h an excitation of the target nucleus. However the d a t a concerning these s t r u c t u r e s were restricted to a limited n u m b e r of systems a n d b o m b a r d i n g energies a n d more systematic studies were necessary to resolve several questions concerning t h e i r nature.
3.
F I R S T E X P E R I M E N T A L RESULTS W I T H AN Ar B E A M AT I N T E R M E D I A T E
ENERGY F r o m D W B A calculations giant resonance cross sections from heavy ion inelastic scattering are expected to be considerably en-
z
4°Ar
9°Zr
8
1000 --3°<8Lob <2.7° ~,~'
2'
1500
/~'
!r :
into t h e i r excitation m e c h a n i s m is to perform e x p e r i m e n t s at higher incident energies. T h e
z
~,
Ein c = L4 MeV/n /,0 50MeV Ar 67 58 I
isoo
h a n c e d at high incident energy• If the high energy s t r u c t u r e s can be related to collective excitations one way to get a deeper insight
+
1250
I"
¢111:
,J,*'*~W,, ,÷~
:
: 1500
5oo-
giant resonances are known to be a general
#~,~*
~ 2.8°<8Lob <32 °
,,*"
¢,.
p r o p e r t y of nuclei• T h u s a b e t t e r knowledge of t h e evolution of these s t r u c t u r e s with mass n u m b e r is required• Therefore we have per-
0:
750
300-~ , ~
,~o~
3.8o
formed e x p e r i m e n t s at GANIL at intermediate energy.
E x (MeV) 70
50
30
Fig. 1. Inelastic 40At + 90Zr spectra obtained at 3 different angles presented in a linear scale with 1 MeV/channel-binning.
N. Frascaria/ Review of high excitation energy structures
247c
A first step was the study of the 4°Ar +40 Ca, 9°Zr, 12°Sn and 2°SPb reactions at 44 M e V / n using a time of flight spectrometer built for these experiments ~. Data were taken for angles close to the grazing angle in each reaction. In fig.(1) are displayed three inelastic 4°Ar +90 Z r spectra obtained at different angles around the grazing angle (Ogr = 2.9°). At angles close to and slightly before the grazing the bumps show up clearly in the spectra up to 60 MeV and they are best observed at forward angles. A very rapid decrease with angle is observed for the structures and for large angles (~ 4 °) they have completely disappeared. These structures have a small cross section and are superimposed on a large physical background. In order to obtain more quantitative information on the high energy structures statistical analyses such as auto-correlation and cross-correlation analyses were carried out on the inelastic energy spectra of the 4°Ar +90 Z r reaction following the methods developed in ref. 8. In these calculations, the structures are analyzed as fluctuations around an average cross section which is defined by the sliding average method. 1
~+~
< a(Ei) >= -2A- + 1 j=i--z~o ( z i ) where A corresponds to an averaging interval of :E 6 MeV in the present case. The auto correlation function is then defined by
C(~)
(a(E + c ) - < a(E + ~) > ~ca(S)-, < a(S) >_~j> < a(E + e) > < a(E) >
In fig. (2) is displayed the result of the auto-correlation calculations for different energy spectra over the excitation energy range 36 < E* < 82 MeV. In such a correlation function the first channel corresponding to the correlation between two unshifted spectra indicates the strength of statistical fluctuations. For small angles the first shoulder observed in the correlation functions demonstrates the presence of structures in the spectra. The strong b u m p centered around e = 9 MeV shows that the experimental structures are regularly spaced by about 9 MeV. Even if the correlation becomes rather weak the presence of the following bump centered around ~ = 20 MeV indicates that at least three structures are present in the studied excitation energy range of the original spectra. At angles larger than about 3.80 where the structures are not seen in the experimental spectra the correlation function exhibits (see fig. (2.1)) a very different pattern. None of the above features is observed. A cross correlation analysis has been carried out following the same method in order to obtain the degree of correlation of the structures observed in the spectra taken at different angles (see fig.(2.2)). The cross correlation analysis was done by shifting the first s p e c t r u m with respect to the second one in order to find the different cross correlation maxima. These maxima are related to different bumps in the spectra and their position gives the average spacing between the bumps. In fig. (2.2) three cross correlation maxima are clearly observed in all cross correlation curves. Again the average spacing between
N. Frascaria / Review o f high excitation energy structures
248c
4°Ar+9OZr E inc =/,4 MeV/n
u \1
o)
Ax
4°Ar
2.3°< 8Lob < 2.8o 0.5 0
U 0.1
U1] n A
n. ~ ] ~ n
hill.
40Ar+90Zr ~ 4 0 A r
.Era.
2.o.4
eLob: 2.7~nd 8Lob =2.~
(-~
0.2 (o)
!
2J~
0
-0.2 -0.5
~0.1
1
b)
3.8°< BLab
eLob:2.8*oncl 8to b :3.2°
-0.40,4~
0.5 0.2 0 --0.2
0.20 -&2
~] 0.5Jl
Honte CarLo
0.1
c)
IU-u.,,"u-u,-', u-u,., ll Uk
0
u
-0.4.
0.2 o
(c]
MonteCorto
JlYJl
-0.1
0
10
Fig. 2.1
20 ~(MeV)30
0,2 o -20
-10
0
10 E.MeV.20( )
Fig. 2.2
Fig. 2.1. Auto-correlation functions for two 40At + 90Zr inelastic spectra and for a random fluctuation spectrum (see ref. 7}. Fig. 2.2. Cross-correlation functions for 40At + 90Zr inelastic spectra a) between eLa b = 2.7 ° and eLa b = 2.8 ° b) between 9La b = 2.8 ° and eLa b = 3.2 ° c) between two random fluctuation spectra (see ref. 7).
N. Frascaria/ Review of high excitation energy structures
249c
these m a x i m a is around 10 MeV which is in agreement with the auto-correlation analysis. Moreover the fact that the cross correlation functions exhibit a m a x i m u m for e = 0 indicates that the structures lie at the same excitation energies, at all angles where they are observed. In conclusion, these analyses quantitatively confirm the existence of physical structures regularly spaced in the excitation energy range of 36 to 82 MeV. As mentioned before, to understand the mechanism of excitation of these physical structures the knowledge of the evolution of the structures with incident energy and with target mass is primordial. A first attempt consisted in the study of the influence of the target. The inelastic scattering of 44 M e V / n 4°Ar from 4°Ca, 9°Zr, l~°Sn and 2°sPb targets has been investigated and the resulting spectra are displayed in fig.(3). Structures are observed in the high excitation energy region in all inelastic channels. These structures are not found at the same excitation energies in the four spectra. The structures seem to be a general property of nuclei and clearly evolve as a function of the target mass. This point will be developed later.
Z'0Ar +90Zr Z
~,
LN.
O O
I00(
Z O O
50C 200 100
60(
400
200 750 500
60( 25(3 108 0
Ex(MeV)80
70
60
50
/~0
30
20
Fig. 3. Inelastic 40Ar spectra from 208pb, 90Zr and 40Ca presented in linear scale with 1MeV/channel binning. (For more details see ref. 7)
Ex (MeV)
70
50
30
Fig. 4. Inelastic 40Ar + 90Zr spectra measured a) at 44 MeV/n b) at 33 MeV/n. Spectrum c) illustrates the effects of the different backgrounds (see ref. 7).
2 50c
N. Frascaria / Review of high excitation energy structures
In these experiments performed with the time of flight system, the influence of the incident energy has also been investigated by studying the 4°Ar +90 Z r reaction at 33 MeV/n.
The comparison of the inelastic channels obtained in the A r ÷ Z r reactions
at 44 MeV and 33 M e V / n is given in fig. (4). At least the bumps at 43, 50 and 58 MeV (+2 MeV) are clearly observed in both spectra while the bump at 66 MeV is less visible in the 33 M e V / n spectrum. This can be an effect due to the rapid fall off of the background in that region (see ref. 7). Thus, for the bumps up to 58 MeV this result is a strong element in favour of a target excitation. But it is well known that three body processes can contribute strongly to the cross section of the studied spectra and consequently before drawing any conclusion a careful study of these processes is necessary. 4. P I C K - U P B R E A K - U P P R O C E S S In this process the projectile picks up one or more nucleons from the target and subsequently decays in flight by the emission of one or several light particles. For example the inelastic spectrum of the 4°Ar+9°Zr reaction will be contaminated by the following two step processes : 40Ar ÷90 Z r
--~ 41Ar, +89 Z r ] ) 4°Ar + n -~ 4aK, +89 y
4OAr +9o Z r
! - -
~ 4°Ar + p
etc...
It is very instructive to calculate 9 by simple kinematical considerations the apparent excitation energy and the kinematical broadening of the bump generated by such a process and to study its dependence on incident energy and on target and projectile masses. This calculation can be done by comparing the two following reactions :
and
1+2 1+2
-~ -~
3~+4 ' 3+4 ]. ) 3 ~÷ 5
As in heavy ion reactions M5 << M4 and M5 << M3 the apparent excitation energy and the width of the pick-up break-up component contributing to the inelastic spectrum can be written : E;p~
MSE
N tab+S.~(4')+ < E 5 > + E r
F Y
F-~4~/
E t?b
M~
N. Frascaria/ Review of high excitation energy structures
251 c
where < E5 > is the average kinetic energy of the emitted light particle in the frame of the projectile-like fragment, $5(4~) is the separation energy of the light particle in the target and E~ is the excitation energy of the target-like nucleus. From these formulae we can deduce that E~:p depends very little on the nature of the target. Thus for a given projectile at a given incident energy, the pick-up break-up contribution is nearly the same for reactions on different targets. The most important term in these formulae is the one proportional to E~b/M3 which implies that E~p and I' depend strongly on the beam energy per nucleon. Thus a variation of the bombarding energy can provide an unambiguous discrimination between target excitation and decay products of excited projectile fragments. This is an important point for the following. In order to describe in detail the energy 40Ar+90Zr spectra of the various transfer evaporation 8 Ein c :/J. MeV/n channels a Monte Carlo calculation has been ~000, 2.3*< 8Lo b < 2.7° performed 9 for the studied reactions using /.0Ar / the code LILITA 10 which assumes a purely y statistical decay process in the framework of the Hauser Feschbach theory where the light particles are emitted isotropically in the cm of the fully accelerated fragment. As
~00 '
-
~
~
the first step of the reaction is not known, this calculation necessitates several assumptions which are discussed in detail in ref 9. Fig. (5) displays the inelastic 4°Ar +~0 Zr spectrum compared to the result from the
~0
LILITA calculation. In this calculation all the evaporation channels have been taken
Fig. 5. Comparison of the inelastic 40At + 90Zr spectrum with the pick-up break-up calculation described in text.
1600
1700ELob(HEY)
into account. The resulting curve is normalized to the experimental spectrum in order to indicate the maximum contribution consistent with the data. From this comparison, one can conclude that : i) the widths of the calculated contributions are much larger than the ones of the observed structures and thus this process contributes only to the background of the spectra ii) the evaporation channels of more than one nucleon contribute to a region where the experimental cross section is very small and where no structures are observed. Before going into the details of the most recent results obtained at GANIL, let me present and discuss results from other laboratories using "light" heavy ion beams. 5. EXPERIMENTAL RESULTS WITH "LIGttT" HEAVY ION PROJECTILES Recently several studies of inelastic scattering of light heavy ions at intermediate
252c
N. Frascaria / Review of high excitation energy structures
energy have been performed. Two of them, the studies of the 2°Ne +208 Pb reaction 1, at 30 M e V / n and of lrO +2o8 Pb at 22 M e V / n 12 concluded to the absence of physical structures in the high excitation energy region. However these conclusions must be taken with caution since in both experiments several structures were observed but attributed to defects in the detection systems and moreover a very recent study 13 of 2°Ne inelastic scattering on 2°Spb has demonstrated that the conclusions of ref. 11 were incorrect and that physical bumps are present in the ZONe ~_~0sPb spectra at 30 MeV/n. Let me now present and discuss the two recent experiments 160 +20s Pb performed at 22 and 25 M e V / n (Oak Ridge) 14 and ZONe +2os pb,9OZr at 25 and 30 M e V / n (MSU) 13 5.1. The 160 + 2°SPb experiment 14
60oo
This reaction was studied at two slightly different incident energies. In fig. 6 are displayed the two inelastic channels from this reaction at 22 and 25 MeV/n. Together
~o0
with the G.Q.R. and two peaks at 13.6 and 17 MeV two bumps are observed at approximately 25 and 40 MeV excitation energies. These two bumps have a cross section only
4o0o
a factor ~ 2 smaller than the G Q R and be-
looo
LILITA as described in paragraph 4 cannot reproduced the two observed bumps. This point will be discussed in section 5.3. 5.2. The 2°Ne + ~°Spb,9°Zr experiment 13
J
~
=
L
~ ~.
~t ill
°
energies are shifted by ~ 2.5 MeV. Clearly these structures cannot be identified with
up break-up component calculated with
,
V
:re0 - 4o0 M,V-2.SI~V~ moo
kinematical evolution predicted for a pickup break-up reaction. Obviously the pick-
,
~ 3o00
tween the two experiments their excitation
high excitation energy states of the target nucleus since the observed shift follows tile
,
~'°spbd60, 160')208pb
0-,2°
l
~ .
.
.
I
t
.
E16 C = 350 MeV
....
~ I
I
I
Oi. b " 14 ° i i I
oss so 4s 4o 3s a0 2s EXCITATION
I
/f
y .
~,~ ~ I
11t
~ -..~J
~ .
~-
ENERGY
20 is
lo
(MeV)
Fig. 6. Inelastic spectra from 160 + 208pb reaction from ref. 14. The solid line represents the calculation of the double humped pick-up break-up component arbitrarily normalized to the data (see text).
The inelastic scattering of 2°Ne on 9°Zr and 2°8Pb has been studied at 500 and 600 MeV incident energies. High statistics spectra were measured at the grazing angle using a magnetic spectrograph. In fig. (7) are displayed the inelastic spectra obtained in these reactions. In the case of the Pb target, structures are clearly observed in the inelastic spectra at the two incident energies but they appear at different excitation energies. A cross correlation analysis of these data was carried out for both studied systems.
In
the case of the Ne + Pb system, the cross correlation function is strongly negative for unshifted spectra, indicating that the structures observed at 600 MeV and 500 MeV incident energies are out of phase (see fig. 8). It takes its m a x i m u m when one of the spectra is shifted by about 5 MeV relative to the other. Again this corresponds exactly
N. Frascarial Review of high excitation energy structures
ZONe + 209pb
253c
20Ne+9°Zr
,x,,
......
)
,=,
/.0000
+,
~
g
+1
g
+,
a)
g 3o0oo ',
>= .i
g "-. m
30000
20000
u o
"" ' Co
IOOOO
E Z
5000
30
40 50 Ex (MeV)
60
30
70
40
50
60
70 Ex {MeV)
Fig. 7. Inelastic scattering spectra at the grazing angle from the reactions 20Ne + 208pb and 20Ne + 90Zr at 500 and 600 MeV incident energies (for more details see ref. 13).
®2.o Ne 6 0 0 M e V
0N e 600MeV 2ONe 5 0 0 M e V
c,,I
2 ° N e 500MeV ( J' -
1
c~
o -I
, ,,
1
0 ,l
,,,,
-4-2024 A (MeV)
-1
I
I
f
,
i
-4-2
i
i
i
i
I
0 2 L, A
(MeV)
Fig. 8. Cross-correlation functions between the spectra taken at 25 MeV/n and 30 MeV/n for the 20Ne + 208pb (a) and 20Ne + 90Zr (b) reactions.
254c
N. Frascaria / Review of high excitation energy structures
to what is expected from a quasi projectile excitation. A different result is obtained for the ZONe +90 Zr inelastic spectra. The correlation function is rather flat and close to zero for unshifted spectra (see fig. 8). Let me now discuss these results in terms of pick-up break-up effects, 5.3. Discussion In the pick-up break-up calculations discussed in the preceding paragraph, continuous excitation energy and spin distributions were postulated for the levels excited just above the emission threshold of the quasi projectile. If now we assume that due to momentum matching conditions a level just above the particle emission threshold is highly excited in the quasi-projectile giving rise to a strong angular momentum alignment the light particles will be emitted following a Legendre Polynomial Pe,~ (cos 0) distribution. A complete calculation 9 shows that the transfer evaporation contribution is then split into two peaks separated by the width F. This double humped contribution follows the same evolution with incident energy and target mass as described before for the flat pick-up break-up component. 3+ L =- 2 level at 5.08 In the case of the 160 +208 Pb reaction, if we assume that the ~ MeV is strongly excited in 1tO with an aligned angular momentum one can reproduce completely the data 9 (fig. (6)).
As already observed in the ~60 +z0s Pb experiment
the shift between the two main bumps observed at 25 MeV/n and 30 MeV/n in the 2°Ne -I-2o8 Pb inelastic channel roughly corresponds to the difference between the two incident energies per nucleon. A complete pick-up break-up calculation shows that these peaks could correspond to the decay of an unbound state in 21Ne which emits a neutron of 0.4 MeV (see ref. 13). One can notice that this phenomenon occurs for quasi projectiles such as 170 and 2lNe having a low level density above the emission threshold. In the case of heavy quasi projectiles such as 41At discussed before, the neutron threshold is high and the level density above this threshold is very large. Consequently in that case the probability to excite preferentially discrete states in the region above the emission threshold is certainly small. But it is not straightforward to appraise the probability of the presence of this double humped structure. It has been shown that, whether double humped structures are present or not, the centroid and the width of the pick-up break-up contribution depend strongly on the beam energy per nucleon. Thus, a clean method to conclude between target excitation and pick-up break-up reactions is to study a given system at two different incident energies.
Even in this case the change of matching conditions
between two very different incident energies could modify the relative population of tile quasi projectile states. Consequently a definite conclusion concerning such a mechanism requires a comparison of two experiments where the bombarding energy per nucleon is varied by approximately half of the mean interval between two observed bumps. 6. S.P.E.G EXPERIMENTS In this paragraph, let me present very recent and preliminary data we have obtained
255c
N. Frascaria / Review o f high excitation energy structures
at GANIL using the magnetic spectrograph SPEG. The aim of this experiment was to compare with high resolution and high statistics the inelastic scattering of 4°Ar on 9°Zr at 41 M e V / n and 44 M e V / n in order to distinguish between excitations of the projectile and the target fragments. The m o m e n t u m range of the spectrograph is sufficiently large so that the entire excitation energy range of interest (~ 100 MeV) could be covered at one magnetic field setting. The total aperture of the spectrograph is 4 ° . The emitted fragments were detected by the standard focal plane detection system consisting of two drift chambers for position measurements and two rnulti-anode ionisation chambers for particle identification.
A time of flight measurement was taken between a large area
parallel plate counter and the cyclotron R.F, An unambiguous mass, atomic and nuclear charge identification of the fragments in the vicinity of 4°Ar was obtained. The first remarkable result of this experiment is the large peak to continuum ratio
GQR i
40Ar + 90Zr
Z.0Ar
"~ "~o 5000-
59 50 66 t ~
4500"
40E I j : : : : M
O
eV/n
4% :
5000-
o
/
LEO~IJ 3000-
~000
¢t/'lt GQR
i
7500
2500-
I~
-5000
]
-2500
2000
20
-8000
ex =2
3000-
2%
8o (..9 12000
~1 ~" ~'
j ~;~#tl~t~#~'
Einc :41Mev/n
Ex (MeV)
..i-,
Ex=36MeV 1/
Einc : ''MeV/n
io
Fig. 9. Low excitation energy region (0 < Ex < 20 MeV) of the inelastically scattered 40At spectrum from 90Zr taken at the grazing angle.
>~
90 70 Ex(MeV)
50
30
Fig. 10. Inelastic 40Ar + 90Zr spectra measured at 41 MeV/n and 44 MeV/n with SPEG with two different representations (see text). The arrows indicate the positions of the bumps observed in the time of flight measurement at 44 MeV/n for the same system.
256c
N. Frascaria / Review o f high excitation energy structures
observed for the giant quadrupole resonance which is considerably e n h a n c e d at t h e studied energy c o m p a r e d to heavy ion e x p e r i m e n t s at lower incident energy (see fig. 9). In fig. (10) are displayed the two inelastic spectra at 44 M e V / n a n d 41 M e V / n incident energies, b o t h w i t h two different energy binnings : 250"keV/channel a n d 500 k e V / c h a n n e l in order to observe more clearly the b r o a d s t r u c t u r e s at high excitation energy. At first glance the s p e c t r a at b o t h incident energies are very similar a n d exhibit s t r u c t u r e s up to ~ 60 M e V excitation energy. In order to insure t h a t these small cross section fluctuations are not due to differential n o n linearities of the position sensitive drift c h a m b e r s m e a s u r e m e n t s were performed with t h r e e different settings of the magnetic field of the s p e c t r o g r a p h at b o t h incident energies. To check t h e consistency of the excitation energy s p e c t r a at different (Bp) cross CROSS CORRELATION
1llIj11111111 CROSS CORRELATION
1.0
^r.Zr 41MeVlu
0.5
0.5
Tlltttr1tttt tt ttttttt
0,0
0.0
tt.o..o
"0.5
t "20
2.BOo3.Bp -10
0 ¢ (MeV}
10
-0.5
20
I
I
-20
I
I
I
I
I
I
-I0
I
I
I
I
I
I
L
I
0 ¢ (MeV)
I
I
I
I
I
~
I
I0
20
Fig. 11. Cross-correlation functions between spectra taken at two different settings of the magnetic field: on the left at 44 MeV/n, on the right at 41 MeV/n. The errors bars are calculated following ref. 8.
correlation analyses were carried out a n d are displayed in fig. (11). T h e fact t h a t the cross correlation functions show a m a x i m u m for eel < ] MeV clearly d e m o n s t r a t e s t h a t t h e m a i n oscillations observed in the s p e c t r a have a physical origin. T h e u n c e r t a i n t y of t h e energy c a l i b r a t i o n is estimated to be less t h a n 1 MeV. Now, we shall c o m p a r e the two inelastic spectra o b t a i n e d at 44 M e V / n a n d 41 M e V / n (fig.
(12)).
These s p e c t r a are preliminary a n d the full statistics have not yet been
analysed. Nevertheless the result is clear : the b u m p s are not shifted a n d show up at t h e same position in b o t h spectra. In the u p p e r p a r t of fig. (12), these two s p e c t r a are s u m m e d for a same n u m b e r of counts a n d in the resulting s p e c t r u m the three observed b u m p s remain. Moreover a cross correlation analysis between the two s p e c t r a taken at 44 M e V / n a n d 41 M e V / n has been carried out. As one can see in fig.
(13)
a strong
m a x i m u m is observed for the unshifted s p e c t r a (~ -- 0) showing t h a t the s t r u c t u r e s are in p h a s e at b o t h incident energies. A s u p p l e m e n t a r y test to show t h e accuracy of this result is given by the c o m p a r i s o n
N. Frascaria / Review of high excitation energy structures
CORRELATION
CROSS 1
>~
>~
•-
.T.
O3 I..-
O3 I--
Z
D O
(.9
1
1
1
1
,
,
,
,
,
l
l
T
J
,
l
J
Ar • Zr 0.5
J
,
44
,
,
,
-
-
MeVIue 41MeVlv
llllllllllIlilllll
z
O
1
257c
0.0
(D -0.5
I I L I I l . L l o l i i i l l l l l l l l i0l ~ o I . 2 0 lO ¢ {HeV}
10 000
7 500
800C
5 000
Fig. 13. Cross-correlation function between the two spectra of the lower part of fig. 12.
Z,OAr +90Zr
o
I.500
450C
>~
t
4OAr
~
o u
6"/50"
#
7°Ex (MeV)5°
4500-
/
/ /z 2250
/
:
.
-5750
(o,/ ~e ' ~ ' J / ,.'
2250-
Fig. 12. Lower part: comparison between the inelastic spectra from the 40Ar + 90Zr reaction measured at 44 MeV/n and 41 MeV/n. Upper part: sum of a 41 MeV/n and 44 MeV/n spectrum having both approximately the same number o f counts.
-9000
/
.4500
(b)/'
./
/
Ex(MeVlO
2250
'/'o
5'0
3'o
Fig. 14. a) target excitation case: same as upper part of fig. 12 with a different energy binning b} quasi-projectile case (see text}.
258c
N. Frascaria/ Review of high excitation energy structures
of the two spectra displayed in fig. 14. Spectrum a) is obtained by summing the spectra at 44 and 41 MeV/nucleon for a same number of counts. The bumps observed in these two spectra are clearly visible in the sum spectrum a) which is expected if they are due to target excitation. On the other hand structures produced by pick-up break-up reactions are expected to shift by 3 MeV between 44 and 41 M e V / n (see paragraph 4). To investigate this last hypothesis, spectrum b) presents the sum of the 44 M e V / n data and the 41 M e V / n spectrum shifted upwards in excitation energy by 3 MeV. Within the statistical error bars spectrum b) is flat showing that the contribution to the structures of double humped pick-up break-up components, if it exists at all, is very weak. Moreover the structureless spectrum obtained in this case shows that the 3 M e V / n energy shift between the two incident energies provides a meaningful test to conclude between target excitations and three body reactions. In fig. (15) are displayed the four inelastic spectra obtained in the A r + Z r reaction
Z,0Ar + 90 Z r
415MeV÷ ÷ + (' Ex:59' ** l ,
4OAr
%
o (J
51, !
l. ,j ;
10000-
:C:in~=4iMelv/n
2co~ (_)
~'
•3600
800( •
#~+!
SPEG:
+
+
.+~, !Einc=4&MeV/n +
600C
2400
.?#
,
)*#,+ )
.#WWTF(x~-) I~ "
r ei~c=41NeV/n .
,~'#I
4500
: Ii,
+
~
,
t
'
J i!
,÷
.~# @'"+, 1200
; I! '
Eihc=33MeV/n
"
#
J
E~'IMe'V) 7'o
'
5'0
'
3'0
'
Fig. 15. Comparison between the inelastic spectra from the 40At + 90Zr reaction measured at 33 MeV/n {time of flight measurement}, 41 MeV/n (SPEG measurement• ) and 44 MeV/n using the two different set ups.
259c
N. Frascaria / Review o f high excitation energy structures
at 33 M e V / n , 41 M e V / n and 44 M e V / n with two different set ups - the time of flight (TF) discussed previously and SPEG. Clearly the bumps show up at the same position at the three different incident energies. One can notice that the two spectra obtained with T F and S P E G are identical and differ only by the number of counts which is a factor 6 larger in the S P E G experiment. 7. D I S C U S S I O N If we recapitulate now all the data obtained with the 4°Ar beam at GANIL it has been shown that i) in the case of the Z r target the positions of all the structures are confirmed with high statistics in the S P E G experiment. This gives confidence in the time of flight measurements presented before ii) the results obtained at 33 M e V / n , 41 M e V / n and 44 M e V / n for the A r + Z r reaction point to an interpretation of the structures in terms of target excitation iii) a clear evolution of the structures with target mass is observed. Thus, from the time of flight data we have tried to extract the different positions of these bumps in the four studied systems. Different methods have been used (see ref. 7) consisting in different background subtractions and a double Fourier transform analysis. A compilation of these results taking into account all the measured angles is shown in fig. (16). The error bars reflect the dispersion on the excitation energies given by the different
A-~'~
/,'LE.O.R /
,,,,o.oi / G.OR
104 (H,ED.R)
/
128 156 12phone //(L=5}
186 (3phonons)
~ /
~(4~onons)
/
-e--~
225 _(L=/,)
~q,
302
~ ~ ~ ~ ( S p h
o 2L
Sn
~
~
..... )
-*/- .+/--.-/~J 1.7~ ~+j
6'0
4 Ex(M~V)~
Fig. 16. Excitation energies of the structures observed in the inelastic channels of the four reactions (40At + 40Ca, 90Zr, 120Sn, 208pb) plotted as a function of target mass A- 1 / 3 Tentative interpretations of the different states are given. analyses and the uncertainties due to statistical fluctuations and energy calibrations. An a t t e m p t was made to reproduce the positions of the structures with different laws of the type E~c~A P but the only consistent representation of the data is given by A -1/3 laws as
260c
N. Frascaria/ Review of high excitation energy structures
presented in fig. (16). In such a representation, the points can be joined with surprising consistency by straight lines passing through 4 data points and the origin. The slopes of these lines in units of M e V / A -1/3 are also indicated. The second slope corresponds to the well known G.Q.R. The first and the third ones can be tentatively associated with the low excitation and the high excitation octupole resonances (LEOR and HEOR). The others are not known and we can try to conjecture what might be the nature of these excitations. To understand the origin of these bumps a large amount of theoretical work has been carried out during these last years 7. Up to now the most consistent representation of the data is given by the multiphonon calculation 6,15 where one supposes the excitation of multiphonon states built with the giant quadrupole resonance. This implies that the multiphonon energies are multiples of the G.Q.R. energy. As one can see in fig. (16) the 128, 186 and 261 M e V / A -1/3 slopes are in surprisingly good agreement with such a picture corresponding respectively to what is predicted for 2, 3 and 4 phonons but all the data cannot be completely interpreted by this model and the nature of these structures remains an open question. 8. CONCLUSION From the presented work, we have learned that projectile effects can be very important in some heavy ion collisions using "light" heavy ion projectiles leading to a highly excited double humped structure in the inelastic spectra. Consequently, if small target excitations are present in the spectrum this effect will not allow to observe them. With a heavy projectile such as 4°Ar, it has been shown that structures are observed in all studied target nuclei and these data are consistent with an excitation of the target. Light particle-heavy fragment coincidence experiments should now provide a unique tool to disentangle the contributions of projectile and target excitations in heavy ion inelastic spectra and will allow a further step towards the understanding of these complex reaction mechanisms.
Acknowledgements : I wish to acknowledge my colleagues D. Beaumel, Y. Blumenfeld, Ph. Chomaz, J.P. Garron, J.C. Roynette, T. Suomij£rvi, J. Barrette, B. Berthier, B. Fernandez, J. Gastebois and W. Mittig for many fruitful discussions and for allowing me to present our unpublished results. Special thanks are due to Y. Blumenfeld for a careful reading of the manuscript. I acknowledge D. Grialou for the elaboration of the manuscript.
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N. Frascaria / Review o f high excitation energy structures
261 c
Tassan-Got, Phys. Rev. Lett. 39(1977)918 2) N. Frascaria, P. Colombani, A. Gamp, J.P. Garron, M. Riou, J.C. Roynette, C. St6phan, A. Ameaume, C. Bizard, M. Louvel, Z. Phys. A294 (1980) 167 N. Frascaria, Proceedings of the Int. Conf. on Nucl. Phys. (Trieste}, Nucl. Phys. C.H. Dasso, R.A. Broglia, A. Winther EDS North Holland, Amsterdam (1982} 617 3) A.C. Mignerey, K.L. Wolf, M. Breuer, B.G. Glagola, V.E. Viola, J.B. Birkelund, D. Hilscher, J.R. Huizenga, W.V. Shroeder, W.W. Wilcke, Proc. of the Int. Conf. on Nucl. Phys., Berkeley, California (1980) 4) S. Pontoppidan, P.R. Christensen, D. Hansen, F. Videback, H.C. Britt, B.H. Erkkiza, Y. Patin, R.H. Stokes, M.P. Webb, R.L. Ferguson, E. Plasil, G.R. Young, Phys. Rev. C 28 (1983) 2299 5) Ph. Chomaz, N. Fra~caria, Y. Blumenfeld, J.P. Garron, J.C. Jacmart, J.C. Roynette, W. Bohne, A. Gamp, W. von Oertzen, M. Buenerd, D. Lebrun and Ph. Martin, Z. Phys. A318 (1984) 41 6) Ph. Chomaz, Y. Blumenfeld, N. Frascaria, J.P. Garron, J.C. Jacmart, J.C. Roynette, W. Bohne, A. Gamp, W. von Oertzen, N.V. Giai and D. Vautherin Z. Phys. A 319 (1984) 167 Ph. Chomaz, th~se de 3~me cycle, IPNO Th 84-01 7) N. Frascaria, Y. Blumenfeld, Ph. Chomaz, J.P. Garron, J.C. Jacmart, J.C. Roynette, T. Suomij~rvi and W. Mittig, Nucl. Phys. A474 (1987) 253 and references therein 8) P.J. Dallimore and I. Hall, Nucl. Phys. 88 (1966) 193 A. Richter, Nuclear Spectroscopy and reactions, Academic Press New York and London, 1974 M.G. Kendall and A. Stuart, Advanced theory of statistics, vol. 3, London (1976) 3rd ed. 9) Y. Blumenfeld, J.C. Roynette, Ph. Chomaz, N. Frascaria, J.P. Garron and J.C. Jacmart, Nucl. Phys. A 445 (1985) 151 10) J. Gomez del Campo and R. Stokstad, Oak Ridge, National Laboratory, report ORNL TM 7295 11) M. Buenerd, J. Chauvin, G. Duhamel, J.Y. Hostachy, D. Lebrun, P. Martin, P.O. Pellegrin, G. Perrin and P. de Saintignon, Phys. Lett. 167B (1986) 379 12) F.E. Bertrand, R.O. Sayer, R.L. Auble, M. Beckerman, J.L. Blankenship, B.L. Burks, M.A.G. Fernandes, C.W. Clover, E.E. Gross, D.J. Horen, J. Gomez del Campo, D. Shapira and H.P. Morsch, Phys. Rev. C35 (1987) 111
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N. Frascaria / Review o f high excitation energy structures
13) S. Fortier, S. Gal~s, S.M. Austin, W. Benenson, G.W. Crawley, C. Djalali, J.H. Lee, J. Van der Plicht and J.S. Winfield, Report IPNO DRE 87-22, Orsay (1987) and to be published in Phys. Rev. C 14) T.P. Sjoreen, F.E. Bertrand, R.L. Auble, E.E. Gross, D.J. Horen, D. Shapira and D.B. Wright, Phys. Rev. C29 (1984) 1370 15) Ph. Chomaz, N.V. Giai and D. Vautherin, Preprint IPNO Th 86-32 Orsay (1986), Nucl. Phys. (to be published) and references therein.