Deep inelastic collisions induced by identical heavy ions

Volume 87B, number 1,2

DEEP INELASTIC

PHYSICS LETTERS

COLLISIONS

INDUCED BY IDENTICAL

22 October 1979

HEAVY IONS

K M HARTMANN Physlk-Department,

Technzsche Vnzversztat Munchen, D-8046 Garchmg, Germany

and W DUNNWEBER Sekrlon Physzk, Vnzversztat Munchen,

D-8046

Garchzng, Germany

Received 23 May 1979

We show that symmetrlzatlon of the doubly dIfferentA cross sectmn d’o/d@dQ, due to the Identity of the proJectlIe and target nuclei, ISable to account for some of the recently observed structures m the energy spectra of products trom the deepmelastlc scattering of 40Ca by 40Ca

In several recent

experimental

studies

of deep-melas-

tic colhslons induced by identical [l-4] and nearly identical [5] nuclei structures m the energy spectra, m addition to the usual [6] quaslelastlc and deep melastic dutnbutlons, have been observed These additional structures were interpreted [l-4] as being due to the excitation of high-energy collective modes [7] and In ref

[5] as being evidence

lar momentum

window

for the existence

of an angu-

for fusion

In this note we study the effect of symmetrlzmg the doubly differential cross section d2u/dBdQ (0 denotes the scattermg angle and Q the energy loss) m the case when there are identical nuclei m the entrance channel Thus cross section may be written as

f@(6) = T(zl+

l)SF((Z)~~(COSO),

(3)

where 11s the angular momentum m the entrance channel, and Sp(Z) the reaction scattering matrix An explicit expression for o(0, Q) has been obtained m refs [8,9] by assuming that the magnitude of S,(l) has a gausslan form m 1 that 1s peaked at Lp with width Ap and that the phase of Sp(2) may be expanded about Lp up to second order -thus mtroducmg the deflection function Op and its derivative 0; The sum over P may be performed [9] by assuming &-function dlstnbu. tlons for Lo, Ap, Op and 0; about their average values LQ, Ap, OQ and Oh m a coarse cell {Q, Q + SQ} We obtain 0(e) Q) = 0(-)(e) Q) + d+)(e) Q) (4)

where for identical spm-zero nuclei m the entrance channel,

7&v =d2 {f,(e) +fpe- 0))

t u(-)(7r - e, (2) + cd+yn - e , Q),

(2)

The sum m eq (1) 1s over all final channels fl which contribute to the energy loss Q for a given exit channel fragmentation In the hehclty representation [8] , the unsymmetrlzed amphtudef&6) (9 = 8 or 71- 0) 1s

where the partial cross sections are given by

and the angular width by EQ = [2/A& +&A;]

1’2

(6) 21

Volume 87B, number 1,2

PHYSICS LETTERS

Table 1 Values of the dynamlcalquantltlesfor the 4°Ca + 4°Ca system Elab (MeV)

Ecm/B

L~az (h)

Lcnt 0~az ~) (deg)

Qmax ~Q (MeV) ~ )

284 400

25 36

94 123

71 78

- 8 7 10 -145 16

28 18 5

In eq (5), k (kQ) is the wave number m the entrance (exit) channel, and PQ is a probablhty density [9], which is constant ff mass transfer is ignored or Q-dependent when It is exphcltly mcluded (see below) In eq (4) we have ignored the interference terms In fact, m a classical friction model, the partxal cross sections o(-)(0, Q) and o(+)(~r - 0, Q) have a negative sign for the polarization of the fragment spins (posltxve polarlzanon being defined as pointing in the direction of the scattering normal k × kQ) while the partml cross sections o(+)(0, Q) and o ( - ) ( r r - 0 , Q) have a positive sign for this polarization [10,11 ] Hence interference can occur only between o(-)(0, Q) and o(+)(lr - 0, Q) and between o(+)(0, Q) and o(-)(Tr - 0 , Q) The interference terms are found to depend critically on the angular momentum L~ Assuming for Lt3, mstead of a 6-function dlstrlbunon, a gausslan distribution of width ×L about the average value LQ, we find that these interference terms are decreased by a factor exp ( - L~,2v21 4 " A-LJ For the examples to be considered presently, values for ×L of 1 or 2 units o f h are suffioent to make the interference structures m o(0, Q) neghglble We perform calculations for the 40Ca + 40Ca system at 284 MeV and 400 MeV [ 1 - 4 ] Since at these energies the ratio of the c m energy to the Coulomb barher (Ecm/B) is much greater than 1 5, negatwe-angle scattering with fusion below some critical angular momentum, Lcnt, is expected to occur [6] The Bass prediction [12] has been used to determine Lcn t The grazing angular momentum Lgra z and the grazing angle 0 . . . . have been calculated using Coulomb trajectories 1/3 + 1/3 a n~l,~ d a r a d m s R =1 225(Aproj Atarg)+20fm The results are gwen in table 1 Several calculations [13,14] and experiments [15] indicate an approximately hnear relationship between the average LQ and the energy loss Q In the light of this we take

LQ = L graz 22

(Lgraz - L cnt)(OJQmax)

(7)

22 October 1979

In accordance with the experimental findings [6] for reactions between medium weight nuclei we choose Qmax = Efoul - Ecm, where ECoul is the Coulomb barrier for rigid spheres For the deflection function we use a form suggested by Ford and Wheeler [16], VlZ,

LQ - Lcn t 2 OQ(ZQ)=Ogra z - p [ln Lgraz_ Lcrltl

(8)

This function vanes slowly for LQ ~ Lgra z and moves rapidly to _~o for LQ ~ Lcrlt - a behavlour of the deflection funcnon exhibited by all calculations where fusion is possible, e g , it closely resembles the deflecnon function calculated by Gross and KalinowsD [ 13] for the Ar + N1 system at 280 MeV The quannty p is deterrmned by taking OQ(Lcn t + 1) = -27r It depends only weakly on the value of ®Q at Lcn t + 1 because of the steepness of ®Q m this region For the various pamal cross sections at 400 MeV the ridge hnes given by IOQ I are displayed m fig 1 Because of the symmetry in the entrance channel there is a ridge line moving for low IQI-values from 0gra z towards 0 (scatterxng through 0) and one moving from rr - 0gra z towards 7r (scattering through 7r - 0) For large IQI-values there is a ridge hne moving from 0 to rr (scattering through - 0 ) and one moving from lr to 0 (scattering through -(~r - 0)) The latter two ridge hnes intersect at 0 = 7r/2 While the symmetnzatlon (2) apphes to all exxt chan. nel fragmentations, the form of the deflection function adopted here is only stated for near symmetric fragmenI

'

I

'

I

4'0

I

'

I

m~- I

~°Ca+ Ca I -O | [ IELAB =/.OOMeV (1'4eV) ~_/o._ ~ ~Q 50 (e,Q)~, o- ('rt;-e, 1~'

L/

130

. I

,

I

LO o

,

I

80 °

-

,

I

,

I

120°eCMI60°

Fig 1 The ridge hnes of the various partial cross sections plotted m the Q-o plane for the 4°Ca + 4°Ca system at Ela b = 400 MeV The assoctated polarizations are indicated by arrows

Volume 87B, number 1,2

PHYSICS LETTERS

tatlon (where the most pronounced structure m the energy spectra is observed [1-4] ) In these exit channels the intensity IS gradually decreased wxth increasing melasticity in the deep-inelastic region due to transfer processes which may be taken Into account by the factor PO in eq (5) Because of the diffusion characteristics of these transfer processes [6]

pQ ~x(1/o) exp [-(Aproj - A e j e c ) 2 / 2 o 2]

22 October 1979 106

(9)

Because of the symmetry m the entrance channel no drift appears in eq (9) We have performed calculations for symmetric fragmentation (A_roj -= for variance o 2 proportional to Q2 [15~, 1 e A elec ) a

PQ ~ 1/Q

(10)

The size of the angular momentum window, AQ, we take, as in ref [9], to be given by AQ = e

dLQ/dQ

(11)

The Q-independent parameter e is obtained by fitting the energy-loss integrated cross section

d~ dO

Q=Omax f

o(0, Q) dQ,

10

(12)

Q=10MeV

at E l i b = 284 MeV [17] and 400 MeV [3] The result-

mg value of At? IS gwen in table 1 The constant of propomonahty in (10) determines the absolute value of the cross section The lower three curves m fig 2 show the energy spectra for Eli b = 400 MeV at angles below, at and above the grazing angle The various pamal cross sectmns contributing to the energy spectrum at 25 ° are also indicated For small IQI-values the partml cross section o(-)(0, Q) dominates Around Q = - 8 5 MeV, oC+)(0, Q) gwes rise to a bump in the spectrum It is noted that due to the large dispersion this bump is centered at less negative Q-values than given by the ridge line (fig 1) For even larger energy losses the partml cross section O(+)0r -- 0, Q) arising from the symmetrizatlon, produces a third bump near Q = - 1 2 0 MeV in the spectrum The posmon of these latter two bumps vanes little with the c m scattering angles considered here [1-4] The top curve in fig 2 is the spectrum at Eli b = 284 MeV and 0era = 30 ° (grazing angle) Comparing the calculated spectra at 284 and 400 MeV, we note that due to the larger energy loss at 400 MeV, the deep inelastic structure is more clearly sepa-

120

80

40 0 -QIMeV)

Fig 2 The energy spectra for 4°Ca + 4°Ca ~ 4°Ca + 4°Ca scattering at Eli b = 284 MeV and Eli b = 400 MeV For the 0cm = 25 ° spectrum the various partial cross sections (r(-)(0 ,Q (short-dashed hne), o(+)(0, Q) (long-dashed hne) and a(+) O r - 0, Q) (dotted hne) are indicated

rated from the quasi-elastic one than in the correspondmg spectrum at 284 MeV The structure appearing at Q ~ - 5 0 MeV at the lower bombarding energy appears at Q ~ - 8 5 MeV at the higher bombarding energy The bump due to the symmetrlzatlon has been pushed from Q ~ - 7 0 MeV to Q ~ - 1 2 0 MeV We have also calculated partial cross sections describing scattering through angles greater than -7r (the m > 0 terms in the Polsson summation formula, see refs [18,19] ) These terms provide a bridge between the deep-inelastic and fusion--fission processes Although they contribute a httle to the magnitude of the energy spectrum m the high Q-tad, they do not introduce any more structure For a direct comparison with the measured spectra one has to take into account the effects of evaporatmn of nucleons and a-particles which may lead to additional structure in the energy spectra of the detected fragments But this effect is more hkely in the region 23

Volume 87B, number 1,2

PHYSICS LETTERS

of thresholds for the first evaporation step and less likely in the high excitation energy region of the third bump Introduced by the symmetrlzatlon The characteristic three-bump structure of our calculated spectra is recognized in the measured spectra [ 1 - 3 ] of near symmetric exit channels in 40Ca + 4°Ca reactions Additional structure in the region of low excitation energies cannot be accounted for by our simple model and signifies structure effects of other origin However, the bump at high excitation energies which is due to the symmetrlzatlon shows up clearly in most of the reported spectra [ 1 - 3 ] It is also Indicated by the strong deviations from a gaussian form of the deepInelastic component visible in the mass-integrated energy spectra of an earlier study [17] of the 40Ca + 4°Ca system In some of the spectra [1 4 ] a larger intensity of the third bump than calculated here IS indicated Fusion-fission processes may account for this difference In a study of the near symmetric system 32S + 27A1 an Indication of two damped components has been found in the energy spectra of various final products [S] We note that for such systems there are always two processes of different net mass transfers with scattering angles 0 and n - 0 leading to the same final products observed at 0 Therefore symmetrIzation effects in the energy spectra similar to those calculated here must be taken into consideration To conclude, the Identity of the projectile and target nuclei leads to observable structure effects in the energy spectra of the resulting products Systematic studies of the doubly differential cross section in partIcular at backward angles and of the polarization (see fig 1) will help to isolate this symmetrlzatlon which is necessary m order to substantiate other effects causing structure in the energy spectra

24

22 October 1979

One of us (K M H ) acknowledges discussions with Y Avishal

References [1] N Frascana et al ,Phys Rev Lett 39 (1977) 918 [2] N Frascaxla et al, Workshop on High resolution heavy 1on physics at 20-100 MeV/A (Saclay, May 31-June 2, 1978) p 72 [3] N Frascana, Proc 5th Aussms Biannual Meeting (March 1979) [4] H Trlcmre et al ,J de Phys 40 (1979) L181 [5] D Doukelhs et al, 7th Intern Workshop on Gross propertxes of nuclei and nuclear excitations (Hlrschegg, Austria, 1979) p 185 [6] J Gahn, Europ Conf on Nuclear physics with heavy ions (Caen, 1976),J de Phys 37(1976)C5-83, L G Moretto, Europ Conf on Nuclear physics with heavy Ions (Caen, 1976), J de Phys 37 (1976) C5-109, and references quoted thereto [7] R A Broglla, C H Dasso and A Wlnther, Phys Lett 61B (1976) 113 [8] V M Strutmsky, Phys Lett 44B (1973) 245 [9] K M Hartmann, Nucl Phys A305 (1978) 279 [10] J Wllczynskl, Phys Lett 47B (1973) 484 [11] W Dunnweber and K M Hartmann, Plays Lett 80B (1978) 23 [12] R Bass, Nucl Phys A231 (1974) 45 [13] D H E Gross and H Kallnowskl, Phys Rep 45 (1978) 175 [14] K Slwek-Wllczynskaand J Wdezynskl, Nucl Phys A264 (1976) 115 [15] WU Sehroder and J R Hmzenga, Ann Rev Nucl Scl 27 (1977) 465 [16] K W Ford and J A Wheeler, Ann Phys 1 (1959) 259 [17] P Colombam et al, Phys Lett 55B (1975) 45 [18] K M Hartmann, Nucl Phys A269 (1976) 237 [19] K Dietrich and Ch Leelerq-Wfllam,Ann Phys 109 (1977) 41