Phase-space constraints on inclusive energy spectra of heavy-ion fragmentation at intermediate energies

Volume 252, number 1

PHYSICS LETTERS B

6 December 1990

Phase-space constraints on inclusive energy spectra of heavy-ion fragmentation at intermediate energies A. Szczurek a n d A. B u d z a n o w s k i Institute of Nuclear Physics, ul. Radzikowskiego 152, PL-31-342 Cracow, Poland Received 10 May 1990; revised manuscript received 1 September 1990

It is shown that the final state density, the so-called phase-space factor, strongly influences the observed energy spectra, leading to a considerable reduction of the width of the fragmentation bump and a shift of the most probable energy at intermediate ( 10100 MeV/A ) energies. This influence, while practically independent of the target, depends on both the projectile and the observed fragment. The effects discussed are consistent with those found experimentally. The bump of multifragmentation is shifted towards lower energies.

1. Introduction At high energy ( > 100 MeV/A) heavy-ion collisions, spectra of projectile-like fragments at forward angles contain a distinct peak with an energy corresponding to the beam velocity and a width corresponding to their momentum distribution in the projectile [ 1,2]. In the intermediate energy range (10100 MeV/A) the width of this fragmentation peak shows a distinct fall-off for decreasing beam energy and an energy shift towards lower energies [3-5]. These features have never been explained satisfactorily [6,7]. It is the aim of this paper to show that the inclusion of the final state phase-space density factor into the expression for the fragmentation cross section leads to the observed reduction of the width of the fragmentation bump and to a shift of its position towards lower energies.

2. Theoretical background

X ~ IT~f12p(K2,~22E,), if

(1)

where Zif is the transition matrix element, p is the

density of the three-body final states (later called the phase-space factor) and vin is the velocity of the relative motion of the particles in the entrance channel. The sum is taken over the magnetic substates of particles in the initial and final states. In the following we assume a quasi-free reaction mechanism i.e. we assume that only part of the projectile r (the so-called participant) interacts with the target producing particles e and R, while the remaining part of the projectile s (the so-called spectator) is undisturbed by the interaction. According to the standard plane-wave impulse approximation (PWIA), the cross section (1) may be expressed by the off-energy-shell cross section for the virtual twobody reaction T(r,e)R, and the momentum distribution of clusters in the composite system [ 8 ] da

Our present analysis is limited to reactions with only three particles in the exit channel. In general, the cross section for a three-body reaction T (P, 12 )3 can be expressed as 18

dtr (27r) 4 _ [ ( 2 S T + I ) ( 2 S p + 1)] -~ dOl dK22dEl hVin

d121 dO2 dEt

d o "°ff-shell

- f~r(p)p(t2~t22E~). dl2e/cM

(2)

The two-body off-energy-shell cross section influences the spectrum of particles e [ 9 ] but has no significance for the spectrum of the spectator particles.

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )

Volume 252, number 1

PHYSICS LETTERS B

In the present paper we consider only the spectrum of the spectator particles, therefore in the following analysis the two-body off-energy-shell cross section will be neglected. According to refs. [ 1,2 ] the momentum distribution of clusters in the projectile is approximately given

6 December 1990

1.2

Et.~=2OOMeV, 7Be

Et.~=l GeV, 7Be

0.8

0.4

~

~

as

f(p) =const..exp(-p2/2t72) .

(3)

o

0

.

.

.

.

.

Etm=2OOMeV, l°B

.

.

.

.

.

.

.

.

.

.

.

E~=IGeV, l°B

It has been shown in ref. [ 2 ] that the following scaling law holds:

a2=a~K(A-K)/(A-1),

(4)

where Kand (A - K ) are the atomic mass numbers of the fragments which constitute the composite system A. In the Fermi gas model ao may be related to the Fermi momentum of nucleons in the composite system by

G~= 1/5P~.

(5)

The parabolic scaling law (4) was confirmed experimentally at high energies with ao approximately equal to 90 MeV/c [ 10 ], which is only slightly smaller than the value deduced for the Fermi gas (100 MeV/c) [21. In the present paper we consider the inclusive measurement T ( P , b ) in which only particle b is recorded. We assume additionally that particle b is a spectator of the quasi-free reaction. An inclusive spectrum of particles b may be obtained by integration of (2) over the directions of the emission of the unobserved particle. It may be shown that the energy spectrum of the outgoing spectator particles does not depend on the angular dependence of the off-energyshell cross section and the integration leads to the formula

d~

d~QbdEb

f dQ~ fb(p)p(~bg2~Eb)

=fb(P) .( d-Q~ p(g2bg2~Eb),

(6)

where the last integral may be calculated analytically (see e.g. ref. [ 11 ] ). The last formula differs from the one that may be obtained when neglecting the phasespace factor

°" 0.0

, ,':

. . . . . . . . . . . . . .

',,,

1O0 200 E (MeV)

i

> . . . . . . . . . . . . . . . . . . .

300

500

1':',, ' 700 900

" ....

11O0

E (MeV)

Fig. l. Some examples of the influence of the density of the final state on the observed spectra for the 5SNi(12C,X) reaction at O= 5 °. The solid line refers to the calculation according to formula (6) (with inclusion of the phase-space factor) and the dashed line to formula (7), The spectra have been normalized to unity.

da

d~b dEb

= CAb

×exp[-Ab(Eb-2~

cos O+Eo)/a 2] ,

(7)

where Ab is the mass of the observed fragment, 0 is the detection angle and Eo= (Ab/Ap)Ep with Ap and Ep being the mass and kinetic energy of the projectile, respectively. The last formula is obtained by a transformation of the gaussian momentum distribution (3) to the LAB system. Usually t7 and Eo in formula (7) are treated as free parameters and are adjusted to experimental data [4,5]. At relativistic energies formula (7) gives a satisfactory agreement with experimental data when using % = 90 MeV/c for various projectiles and targets [10] and when Eo is slightly decreased as compared with the kinematical predictions.

3. Results and conclusions

We have calculated the energy spectra of different 19

Volume 252, number 1

PHYSICS LETTERS B

6 December 1990

20

~Ni(~C,X)

~Ni(12C,X) A: :A: :~"A 7 ~

15

~80

A, ®

O+ A - - * * - m ~ - + A**@~ e~

~'~_]

11B

.~ +A~®

"-;;-'~

®

:4:::

~60 -I

7Be

5

__t9~__11B

40

. . . . . . . . |, . . . . . . . . h . . , , , , , . | . , , , , , , . , i

......... |,*,,,,,,,

>~'90

~Zn(4OAr,X)

ww

~ .,,,_1 Hll llllla

A **:e®~' - |

JE

~ °~8 0

:.:.:..-

/5 >

**@OA~a~O~OO~lIuew + ~AAAAAAAAAA

: : : : : 2 _4~lg **~ 31P *****

"O

°70

° . . . . . . . |,,,,o0.,,| . . . . . . ,,,e . . . . . . . . . h . . . . . . . . e. . . . . . . . I

~Zn(4aAr, X)

A e

5

"O

++++++++++++++++++ DOODODDOODODODDO~OD

®

60

........

I,1111,,1111,,1

:i ++"

nOD

+

0

r"

"~60 O

I .........

I ........

,,*,,,,,I ......... h . , , , , . , , I ....... ,*h,, ...... h°***,,,,

i1,,,1,,,,,

5eNi(X,1°B)

/5

~80 o~

.....

~Ni(X,~°B) o

..a"""""

• J'J'J-' 12(:: ::::: 160

n

>

~,10

O n

+

ODDnQrlO~DrsoDDn +

O

. . . d ~ . 12(; : ; : 1 ; 180 ,~u~u~ 2(~Me

+ O

***** 24M9 [] []

O A+

D

A

5

D

+ ++++++t"+++++++++

AAAAAAAAAAAAAAAAAA

, , , , , , , , h , ....... h.,,,,,,°l ......... ,,.,*,*,,,1 ....... ,, ......

..... :i'5 .....

'i5 .....

E/A

20

40 60 80 100 120 E/A (MeV/amu)

Fig. 2. R e d u c t i o n o f t h e w i d t h o f t h e f r a g m e n t a t i o n b u m p (left p a n e l ) a n d t h e shift o f t h e e n e r g y w h e r e t h e c r o s s s e c t i o n t a k e s its m a x i m u m ( f i g h t p a n e l ) for d i f f e r e n t ejectiles in t w o r e a c t i o n s , a n d for a s e l e c t e d ejectile for v a r i o u s projectiles: ( I ) 5 8 N i ( ] z c , x ) ( O = 5 ° ), ( I I ) 6 8 Z n ( 4 ° A r , X ) 0 = 5 ° ), ( I I I ) 5 8 N i ( X , I ° B ) ( 0 = 5 ° )

fragments for different projectile-target combinations according to formulae (6) and (7). The momentum distribution function (3) was taken, with tr given by the Goldhaber rule (4). The parameter ao was taken as 90 MeV/c, a value appropriate for relativistic energies. 20

Some examples of the influence of the phase-space factor on the observed spectra are shown in fig. 1. It is clearly seen that at lower energies the phase-space factor strongly influences the high-energy side of the fragmentation bump. This influence is the strongest for particles with masses close to the projectile mass;

Volume 252, n u m b e r 1

PHYSICS LETTERS B

for light fragments the influence is found to be smaller. The inclusion of the phase-space factor causes an effective narrowing of the fragmentation bump and shifts the maximum towards lower energies. In fig. 2 we present some examples of the reduction of the width of the fragmentation b u m p and of the shift of the energy where the cross section takes a maximum as compared to the standard approach. We present the "observed" reduced widths. The "observed" reduced widths have been calculated in the following approximate way: (%~bs m 0 " 0 , / / 2 / , / / 1

6 December 1990

100

80 .--

...... O~ • **** 60

40o

......

.....

.....

i(b)

4°Ca(4°Ar,~'S)

3O

(8b)

where Ni(E) is the energy spectrum of a given fragment in the LAB system and the integration extends over the whole fragmentation bump. The index i= 1 corresponds to formula (7) and i = 2 to formula (6). E mi a x is the energy where the cross section takes its maximal value. The reduction turned out to be dependent on the mass of the observed fragment. The heavier is the fragment, the larger the reduction is. This effect is practically independent of the target (not presented in the figures). The effect, however, strongly depends on the projectile (see fig. 2). The predicted influence is consistent with experimental evidence, see, e.g., refs. [ 4,5 ]. IfE~a~ in formula (8) is replaced by an average energy of the fragmentation bump, , the effect of reduction is even larger. If the particles accompanying the measured spectator are excited, which is highly probable at intermediate energies, then the phase-space factor may influence the fragmentation bump even more strongly. An example of such an effect is presented in fig. 3. We present both the reduction of the width (a) and the shift of the most probable energy (b). The present status of experimental knowledge is the following. All energy spectra measured at intermediate energies exhibit features similar to those obtained in the present paper. They are peaked at an energy smaller (by a few to several MeV) than that corresponding to the beam velocity and exhibit a larger tail on the low-energy side of the fragmentation bump.

.....

gev

E/A (MeV/omu)

(8a)

and the energy shift directly as 2. . . .

"1" A A II1"@ "1" A +A * o "1" d ~ ~ ,I-&~ 0

D

0

U "U

40

dE = E m la x - E

4aCe(4°Ar,36S)

~ 6o

,

A2 = f [ (E-Wmax)2Ni(E) ]dE

(a}

>

...... ::::: ~'~'~ *****

0M~/ 20 MeV 4 0 MeV (g) MeV

A

~-20

A

~k

A

r"

10

÷

++++++'*''lillill! AAA**

~ ,ib

OI:IDOOODDDOI~

00r"""'"h"ll''lllP20 40...... 60'"'"''"'"80 ......... I00'""''"'120

E/A (MeV/amu) Fig. 3. Influence of the excitation energy of the unobserved particles on the degree of reduction in the width of the fragmentation b u m p (a) and on the shift of the energy where the cross section takes its m a x i m u m (b) for 4°Ca(4°Ar, 36S) at 0 = 5 °.

This is always ascribed to the energy dissipation mechanisms. However, in may cases the low-energy side of the fragmentation bump may be described by parametrization (7) with ao taken from relativistic collisions (see e.g. ref. [5], figs. 5 and 7b). In the light of the present paper the effect of the asymmetry of the fragmentation bump may be explained by the damping of the high-energy side. In ref. [ 4 ], where the fragmentation of 4°Ar was analysed, ao was found to be strongly dependent on the mass on the fragment. For the S and C1 isotopes the value was found to be reduced the most. It is nicely represented in our fig. 2. Our present analysis has been limited to the break21

Volume 252, number 1

1.2

PHYSICS LETTERS B

4OCo(4OAr,31 p )

1.0

.... ~.~

),~'%

o.,

I1"

0.6

\ X;

III,

o.,

. . . . .

"~00-

/;,::'

. . . . . . .

700

900

1100

E (M~V) Fig. 4. Energy spectrum of the 31p fragments from the multifragmentation of 4°Ar into 31p + 7Li + 2n (five-body) compared with the spectrum of the fragmentation of 4°Ar into 3lp+9Li calculated according to formula (6) (three-body) and (7) (standard). The spectra, normalized to unity, were calculated for the 4°Ca target at a beam energy 27 MeV/A at 0 = 5 °.

up of the projectile into two particles. It is obvious that in reality the emission of more particles may occur leading to additional constraints on the available phase space. One may find some indications in the literature that in the case when more than three particles in the exit channel are favourable, since the particle complementary to the measured one is unstable, the fragmentation bump is strongly modified. The modification is connected with an enormous (about 100 MeV) shift of the fragmentation bump with respect to the position corresponding to the beam velocity (see, e.g., ref. [5], figs. 4, 7a; ref. [12], fig. 6). In fig. 4 we present an example of a spectrum for the 4°Ca(4°Ar,31p) reaction (at the beam energy 27 MeV/A, O= 5 ° ), assuming a multifragmentation of 4°Ar (into 3~p+TLi+2n). In analogy with formula (2) the transition probability was assumed to be proportional to the momentum distribution of the spectator in the projectile. The many-body phase-space factor was calculated by the Monte Carlo method ac-

22

6 December 1990

cording to ref. [ 13 ]. Additionally, we have presented a spectrum of the break-up of 4°Ar into 31p+ 9Li calculated with formulae (6) and (7). One may see a pronounced shift of the fragmentation bump with respect to the predictions of formula (6). For comparison, see the experimental spectrum from fig. 4 in ref. [5]. We believe that in spite of some simplifications in the present calculations the general tendencies of the modification of the energy spectra by the phase-space factor are clearly visible. In conclusion, we suggest that most analyses of heavy-ion projectile fragmentation at intermediate energies have to be reinvestigated including the phasespace factor, and we expect that many of the results will have to be revised. The width of the momentum distribution obtained when neglecting the phase-space factor may be significantly biased and therefore a wrong conclusion on the reaction mechanism may be drawn.

References [ 1 ] H. Feshbach and K. Huang, Phys. Lett. B 47 (1973) 300. [2] A.S. Goldhaber, Phys. Lett. B 53 (1974) 306. [3] F.G. Stokstad, Comm. Nucl. Part. Phys. 13 (1984) 231. [4] F. Rami, J.P. Coffin, G. Guillaume, B. Heusch, P. Wagner, A. Fahli and P. Fintz, Nucl. Phys. A 444 (1985) 325. [ 5 ] Y. Blumenfeld, Ph. Chomaz, N. Frascaria, J.P. Garron, J.C. Jacmart, J.C. Roynene, D. Ardouin and W. Mittig, Nucl. Phys. A 455 (1986) 357. [6] W.A. Friedman, Phys. Rev. C 27 (1983) 569. [ 7 ] A. Bonasera, M. di Toro and C. Gr6goire, Nucl. Phys. A 463 (1987) 653. [8] I. ~laus, R.G. Allas, L.A. Beach, R.O. Bondelid, E.L. Petersen, J.M. Lambert, P.A. Treado and R.A. Moyle, Nucl. Phys. A 286 (1977) 67. [ 9 ] A. Szczurek, A. Siwek and A. Budzanowski, Z. Phys. A, to be published. [ 10] D.E. Greiner, P.J. Lindstrom, H.H. Heckmann, B. Cork and F.S. Bieser, Phys. Rev. Lett. 35 (1975) 152. [ 11 ] G.C. Ohlsen, Nucl. Instrum. Methods 37 (1965) 240. [ 12 ] V. Borrel, B. Gatty, D. Guerreau, J. Galin and D. Jacquet, Z. Phys. A 324 (1986) 205. [ 13 ] F. James, CERN Computer Program Library, W505 (long write-up) (1970).