**Study of peripheral heavy-ion collisions**

**by residue-particle angular correlations**

A. ITALIANO(1), A. TRIFIRO` (1), G. PISENT(2) and A. STRAZZERI(3)
(1_{) INFN, Gruppo Collegato di Messina - Messina, Italy}

*Dipartimento di Fisica dell’Università - Messina, Italy*
(2_{) Dipartimento di Fisica dell’Università - Padova, Italy}

*INFN, Sezione di Padova - Padova, Italy*

(3_{) Dipartimento di Fisica dell’Università - Catania, Italy}*INFN, Sezione di Catania - Catania, Italy*

(ricevuto il 2 Gennaio 1997; approvato il 18 Agosto 1997)

**Summary. — A closed-form expression, which describes in a single picture the**

nonequilibrium component and the evaporation component of the angular correlation
between particles and reaction residues emitted in a peripheral heavy-ion collision, is
derived in the hypothesis of sequential process. As an example, the application to the
*C-a* differential multiplicity for the deep inelastic collision 16_{O1}58_{Ni} _{at}
6 MeV/nucleon is quoted.

PACS 24.90 – Other topics in nuclear reactions: general.

**1. – Introduction**

In several studies of heavy-ion reactions, at incident energies not exceeding about 20 MeV/nucleon and for projectile-target combinations forming a nuclear system for which binary fission is weak, evidence for the emission of particles and light clusters from the continuous part of the spectra of the ejectiles has been shown [1].

Many features of these emissions, investigated by angular correlation methods, are explained in terms of break-up of the projectile and of emission of particles from reaction residues [2-4]. The experimental observations have stimulated many theoretical models and approaches [5-7].

For peripheral collisions, in which two fragments with charge and mass close to the projectile and target are emitted, associated with a few particles, the observed energy and angular correlations between these particles and the reaction fragments have been understood in terms of sequential emission from the detected projectile-like and the undetected target-like fragments. The significant fraction of the yield of the highly energetic charged particles in the forward hemisphere, not explained by statistical emission from accelerated fragments, indicates the presence of an important non-equilibrium component in the particle emission. In particular a common feature in

these coincidence measurements is the observation of a double forward-peaked structure with a minimum near the direction of the detected projectile-like fragment and a marked difference between the emission probability for positive and negative angles [2,4]. These observed features have been described in terms of a theoretical approach [7] which accommodates in a simple way the nonequilibrium component together with the evaporation component of the sequential particle emission in peripheral heavy-ion collisions.

In this paper, following closely the treatment given in ref. [7], we derive a closed-form expression for the b-c multiplicity of a sequential process of the type A( a , b) B( c ) C and we outline that even in the case of a sequential process there is an important and remarkable nonequilibrium component in the particle emission and how useful information on the mechanism of a peripheral collision A( a , b) B can be obtained from the investigation of the b-c angular correlation pattern around the forward angles.

*In conclusion we consider the application to the C-a differential multiplicity for the*
deeply inelastic collision16

O158_{Ni at 96 MeV laboratory energy [3,4].}

**2. – Particle-particle angular correlation**

We consider a sequential process of the type A(a, b) B(c) C assuming that it proceeds
*through a given continuum state (e **B*, J*B*p*B) in the nucleus B to a narrow definite state

*(e **C*, J*C*p*C) in the final nucleus C.

*From now on, e **X*indicates the excitation energy of the state of definite spin J*Xand

*parity p*X *in the nucleus X and m*X * the z-component of J*X. The pair (xX) has relative

**radial coordinate r**x**, momentum k**x**, velocity v**x *and energy e*x. The spherical polar

*angles (w*b*, W*b**) of k**b **are defined in the ( A 1a ) centre-of-mass (c.m.) system, while k**c

*has polar angles (w , W) defined in the recoil centre-of-mass (r.c.m.) system (rest frame*
*of the nucleus B) and described in a xyz-frame with the x-axis and z-axis parallel to the*

*x-axis and z-axis of the c.m. frame.*

We require, for the A( a , b) B( c ) C reaction to be a sequential process, that the
*excitation energy e **B of the intermediate system B formed in the first step of the

three-body reaction be independent of the angle of the particle c and we also assume that in the decay B Kc1C the nuclear interaction between b and B is negligible.

For simplicity, we suppose that the nuclei A , a , b and c have spin zero and b and c are in the ground state.

**2**.*1. Preparation of the intermediate system. – The outgoing part of the stationary*
*solution of the Schrödinger equation for the reaction A( a , b) B[e **B] corresponding to

*the partition ( bB ; e **B*, J*B*m*B*), associated with the unit flux in the partition ( aA ; e*a), in

*the asymptotic region is given by [8, 9] (v f (w , W))*

Cb*4 f*ba*(e **B*; m*B*, v*b*) f*b*f*BOb.

(1)

Here the outgoing wave Obis given by

Ob*4 (v*b1 /2*r*b)21*exp [i(k*b*r*b*2 h*b*ln 2 k*b*r*b*1 s*0b) ]

(2)

*with h*b *and s*0b *the Coulomb field parameter and the S-wave Coulomb phase shift for*

*and B, respectively; the amplitude f*ba, related to the differential cross-section for the
reaction A( a , b) B by means of
*ds*
*dv*b
4

### !

*B*

_{m}*Nf*ba

*(e **B

*; m*B

*, v*b) N2, (3)

*determines the population of the substate (e **B*; J*B*m*B), in the nucleus B, prepared by

**detecting the particle b in the fixed direction of k**b.

**2**.*2. Angular correlation formula. – Now we assume that, outside of the range of*
any interaction with the nucleus b, the nucleus B decays into c 1C. In the rest frame of
the nucleus B, the outgoing part of the stationary Schrödinger equation corresponding
*to the decay from a substate (e **B*; J*B*m*B*) to a given partition ( cC ; e **C*, J*C*m*C), in the

asymptotic region can be expressed by [10]

*f*B*4 F*Bc*(e **B*, e **C*; m*B*, m*C*, v) f*c*f*COc,

(4)

where the function Oc *is expressed in a form analogous to eq. (2), f*c *and f*Care the

normalized wave functions of the nuclei c and C, respectively, and the decay amplitude

*F*Bcis defined by
(5) *F*Bc*(e **B*, e **C*; m*B*, m*C*, v) 42 i*

### !

*l*(2)

*l*C

_{aJ}*lm*C

*, m*B

*2 m*C

*NJ*B

*m*Bb Q Q S

*l(e **B

*, e **C

*); Ylm*B

*2 m*C

*(v).*

*In eq. (5) l is the relative orbital angular momentum of the pair (cC), Ym*

*l* *(v) are the*
spherical harmonics, *aJ*C*lm*C*, m*B*2 m*C*NJ*B*m*Bb are the Clebsch-Gordan (CG)

coefficients [11], S is the matrix which describes the decay B Kc1C and the
*summation over l is restricted to only parity-conserving values. Note that, as f*B in

eq. (1), wave function (4) is normalized. In fact, by using the ortonormality properties
of the spherical harmonics and of the CG coefficients, by integrating over the internal
*variable t of the pair (cC), since the functions f*c *and f*C are normalized and the S

matrix is unitary [9], from eqs. (4) and (5) we have

### !

*m*B*m*C

_{de *}_{B}

_{de *}_{C}

_{dv}

_{dtNf}_{B}

_{N}2

4 1 . (6)

Now in order to evaluate, in the rest frame of the nucleus B, the outgoing part of the
wave function which describes in the asymptotic region the sequential decay process
*( aA ; e*a*) K ( bB; e**B*J*B*) K ( bcC; e**C*J*C*m*C*) we can substitute the wave function f*B

appearing in eq. (1) by the wave function given by eq. (4) and, in the rest frame of the
nucleus B, find
Cbc4 Fbc*(e **B*, e **C*; m*C*, v*b*, v) O*bOc*f*b*f*c*f*C,
(7)
where
Fbc*(e **B*, e **C*; m*C*, v*b*, v) 4*

### !

*m*B

*I f*ba

*(e **B

*, m*B

*, v*b

*) F*Bc

*(e **B

*, e **C

*; m*B

*, m*C

*, v)*(8)

with I the Jacobian of the trasformation from the c.m. system to the r.c.m. system [9]
*(see fig. 1). From eq. (7), the number of the nuclei c emitted in the solid angle dv along*

Fig. 1. – Schematic diagram, by assuming the sequential process A( a , b) B( c ) C, to define the
*in-plane angle W and the solid angles v*b *and v, corresponding to the hatched angular sectors,*
*subtended by the angle-fixed b-particle detector D*b *and angle-variable c-particle detector D*c,
respectively. Labels L and cm refer to the laboratory and ( A 1a ) centre-of-mass system,
**respectively. The circle is centered at v**L

B**with the radius equal to the velocity v**cof the particle c in
**the rest frame of the decaying fragment B. V denotes the velocity of the cm-system in the**
laboratory system (see the text).

*given by the flux j*bc*in the partition ( bcC ; e **C*J*C*m*C) through the surface elements of

*area r*2

b*dv*b *and r*c2*dv, respectively, and defined by [8, 9]*

*j*bc*(m*C*; v*b*, v) 4v*b*r*b2*dv*b*v*c*r*c2*dv*

*dzNC*bcN2,

(9)

where the integration is taken over the totality of the internal variables.

*If we recall the unit flux in the partition ( aA ; e*a*), j*bcis just the double differential

cross-section d2

*s for the sequential reaction ( aA; e*a*) K ( bB; e**B*J*B*) K ( bcC; e**C*J*C*m*C).

*So, by performing the integration in eq. (9), with the functions f*b*, f*c *and f*C

normal-ized, the double differential cross-section for any final spin, from eqs. (7)-(9), is
d2_{s}*dv*b*dv*
4

### !

*C*

_{m}### N

### !

*m*B

*F*ba

*(e **B

*; m*B

*, v*b

*) F*Bc

*(e **B

*, e **C

*; m*B

*, m*C

*, v)*

### N

2 , (10) where*F*ba

*(e **B

*; m*B

*, v*b

*) f I f*ba

*(e **B

*; m*B

*, v*b) . (11)

A quantity often considered in the b-c coincidence measurements is the b-c differential multiplicity defined by (see eqs. (3) and (10))

*M(v) 4* d
2_{s}*dv*b*dv*

### N

*ds*

*dv*b , (12)

that is the number of particles c per nucleus b and per unit solid angle in the moving
frame of the decaying nucleus B. It has the dimensions sr21_{.}

**3. – Residue-particle angular correlation**

The coincidence cross-section so far obtained refers to specific energies of the
nuclei B and C. The theoretical expression of the b-c angular correlation for the
three-body sequential process A( a , b) B( c ) C presented in this section will be obtained
under the hypothesis that the nucleus B is highly excited to levels that lie in the
*continuum region, within an energy interval D centered at the average energy e **B, and

the nucleus C is left in any of a large number of states contained within an interval
*D8AD centered at the average energy e**C. For simplicity, in the following we shall

*consider only the case of a single J*B-value.

So in practice the experimental angular correlation is the sum of double differential
cross-sections, as given by eq. (10), corresponding to excitation energies being in
*regions about the average energies e **B*and e **C, respectively.

Firstly we shall take the average of the double differential cross-section, given by
*eq. (10), over an energy interval D centered at the energy e **B with the following

*definition of the average of a function f (e) taken over an interval D centered at e*0

*a f(e*0)bD4

*de r(e , e*0*, D) f (e) .*

(13)

Note that the final result of averaging should not depend in general on the exact form
*of the weighting function r(e , e*0, D), for the result to have physical sense.

*Besides we assume that the amplitude f*ba (and hence the differential cross-section

*for the reaction A( a , b) B) is a slowly varying function of the excitation energy e **B

within the region of interest D, so that we can get (in what follows we shall omit the
*explicit indication of e **B*, e **Cand D)

### (

see eq. (3)### )

### o

*ds*

*dv*b

### p

C### !

*B*

_{m}*Nf*ba

*(m*B

*, v*b) N2 (14)

and write

### (

see eqs. (5) and (11)### )

### o

d2

_{s}*dv*b

*dv*

### p

C### !

*J*C

*m*C

### N

### !

*m*B

*F*ba

*(m*B

*, v*b

*)aF*Bc

*(m*B

*, m*C

*, v)b*

### N

2 . (15)**3**.*1. Average angular correlation. – In order to average the value of the b-c angular*
*correlation over the interval D centered at e **B, we split the matrix S

### (

see eq. (5)### )

into anequilibrium (E) and a nonequilibrium (NE) terms as [12] S 4SE

1 SNE
*(16a)*

with
SE
4 S 2 aS b ,
*(16b)*
SNE
4 aS b
*(16c)*

and we suppose the phase of SE _{and S}NE _{uncorrelated (so that their cross terms}

average out to zero) and the statistical assumption that in the energy interval D
*around e **B there are many levels contributing to the decay B Kc1C and that their

widths and energies are randomly distributed so that interference terms generally vanish [8, 13].

We also assume that the amplitude SNE

_{(}

_{see eq. (16c)}_{)}

_{is a very smoothly varying}

*function of the excitation energy e **Cwithin a region D8(AD).

Now, it should be noted that the energy fluctuating amplitude SE

_{(}

_{see eq. (16b)}_{)}

_{,}

corresponding to a situation in which the number of various possible modes of decay
B Kc1C is large, varies rapidly also over an energy interval of order of the typical
*width of the many levels contained in the interval D8(AD) around e**C. So that, since the

*number of final states is large and the excitation energies are high, the e **B-energy

*averaged equilibrium angular correlation ( ds)*E_{, described by S}E_{, must be subjected}

*also to an e **C-energy averaging process

### (

see eq. (13)### )

to yield a smoothly varying*function of energy e **C.

*In line with the above arguments, the average of ( ds)*E_{in the interval D8 centered at}

*e **Ccan be approximately given in terms of just the quantity ( d2*s)*E, associated with the

*mean value e **C*, times the relative density of states of angular momentum J*C at the

*excitation energy e **Cin the nucleus C [8, 13].

Finally, if the above restrictions and approximations can be justified, the energy-averaged b-c angular correlation, given by eq. (15), from eqs. (5), (10), (11), (13) and (16), can be expressed as

### o

d2

_{s}*dv*b

*dv*

### p

4## g

d 2

_{s}*dv*b

*dv*

## h

E 1## g

d 2

_{s}*dv*b

*dv*

## h

NE (17) with## g

d2*s*

*dv*b

*dv*

## h

E 4### !

*m*C

### !

*lJ*C

*wl(J*C)

## g

*Tl*

*G*

## h

### N

### !

*m*B

*pl(m*B

*, m*C

*; v*b

*, v)*

### N

2 , (18)## g

d2

_{s}*dv*b

*dv*

## h

NE 4### !

*m*C

### N

### !

*lJ*C aS

*l*b

### !

*m*B

*pl(m*B

*, m*C

*; v*b

*, v)*

### N

2 , (19) where (20)*pl(m*B

*, m*C

*; v*b

*, v) f (2)lF*ba

*(m*B

*, v*b

*)alJ*C

*, m*B

*2 m*C

*, m*C

*NJ*B

*m*B

*b Ylm*B

*2 m*C

*(v) .*

*In eq. (18) the quantity wl*, related to the relative density of the available states

*(e **C

*, J*C

*p*C

*) in the nucleus C, describes the probability of orbital angular momentum l*

transferred in the decay ( *B ; J*B*e **B*) K ( cC; lJ*C*e **C) and we have assumed the

parametrization aNSEN2*b 4Tl/G, where Tl*is the optical-model transmission coefficient
*and G represents all decay modes energetically open for the decay B Kc1C [8, 13].*

It should be noted that the two terms on the right-hand side of eq. (17) are physically different in nature.

Actually, by using the time-dependent scattering theory [14], it can be roughly
assumed that the quantity ( d2*s)*NE is associated with a situation in which the
dissociation of B into c and C is a fast process occurring in time scales by many orders
of magnitude shorter than the typical time corresponding to the equilibrium decay
process, described by ( d2

*s)*E, whose long lifetime leads to the “loss of memory” of the
formation of the decaying nucleus B [13]. In this sense the angular symmetry of the
c-emission from a statistical equilibrated system described by the b-c angular
correlation (18) cannot serve as evidence for any particular model of dynamical
effect.

On the contrary what we can see from the b-c angular correlation (19) is that the memory of the first step of the sequential process A( a , b) B( c ) C can be retained during the subsequent “fast” decay B Kc1C, so that the angular dependence of the particles c emerging from such a short-lived composite system can display a marked forward-backward asymmetry about the direction of the coincident projectile residue b or the beam axis.

Thus the study of the nonequilibrium sequential component of the particle emission is especially useful to probe the early stage of the peripheral collision and can be an useful alternative technique to obtain reaction mechanism information complementary to the ones extracted from the ordinary angular distributions of the two-body reaction products.

**3**.*2. Information on reaction mechanism. – In the case where the interest in using*
the angular correlation method is mainly to obtain information on the mechanism of the
reaction A( a , b) B and on the polarization effects of the nucleus B it is convenient to
* choose coordinate axes so that the z-axis is along k*b

**3 k**

*a*(perpendicular to the reaction

*. Note that the sign convention is here chosen so that a*

**plane) and the x-axis along k**a*c*

**positive in-plane angle W corresponds to an angle of k****with respect to k**

*a*in the reaction

**plane, such that k**

*a*

**3 k**c

*D 0. So that we have W*bE 0. Note also that our convention is

different from the usual Madison one [15].

We shall consider the effects of some approximations on the various quantities appearing in eq. (20).

*Firstly, we can represent the amplitude F*ba*(m*B*, v*b)

### (

see eqs. (3) and (11)### )

forconvenience in the form

*F*ba*(m*B*, v*b*) CNF*ba*(m*B*, v*b*) Nexp [2ijm*B] .

(21)

Besides, since in peripheral heavy-ion collisions there is experimental evidence that the outgoing residues have intrinsic angular momentum predominantly oriented perpendicular to the reaction plane [16], we shall admit that

### (

see eqs. (11) and (14)### )

*NF*ba*(m*B*, v*b*) NCNF*ba*(Nm*B*N 4 m*0*, v*b) N

(22)

*with m*0**the component of J**B*along the z-axis. In the processes here considered J*Band

*J*C*are usually large as compared to l so that we can replace the CG coefficients by their*

asymptotic expressions [11]

*aJ*C*lm*C*, m*B*2 m*C*NJ*B*m*B*b 4dmml* (L)
*(23a)*

*where dl*

*mmis the reduced rotation matrix with m f m*B*2 m*C*, m f J*B*2 J*Cand L given by

L 42cos21

## g

*m*B

*J*B

## h

.
*(23b)*

Then, if the above approximations and restrictions can be justified, from eqs. (21)-(23)
*we have (m f m*B*2 m*C*, m f J*B*2 J*C)

(24) *pl(m*B*, m*C*; v*b*, v) C (2)lNF*ba*(Nm*B*N 4 m*0*, v*b*) Nexp [2ijm*C] ) Q

*Q Dmml* *(jL0 ) Ylm(v) ,*
*where Dl*

*mm* is the rotation matrix corresponding to the rotation of axes through the
*Euler angles (j , L , 0) and we used the property [11]*

*Dl*

*mm(jL0 ) 4exp [2ijm] dmml* (L) .
(25)

Finally, we assume that in the decay B Kc1C the emission of the particle c does not
**change the sign of initial polarization so that J**C**is in the same direction of l.**

Then, by using the transformation [11]

*Ylm*(U , F) 4

### !

*mD*

*l*

*mm(jL0 ) Ylm(w , W) ,*
(26)

the energy-averaged b-c differential multiplicity, describing the angular distribution of the relative probability of the c-emission in the rest frame of the nucleus B, from eqs. (12), (14), (17)-(19), (24) and (26), is given by

*M(v) 4*

### (

*M(v)*

### )

E 1### (

*M(v)*

### )

NE*(27a)*with

*(27b) (M(v) )*E

_{4}

### !

*lm*

*wl(m)*

## g

*Tl*

*G*

## h

### (

*Y*

*m*

*l*(U , 0 )

### )

2 ,*(27c)*

### (

*M(v)*

### )

NE*4 M*0

### m

### N

### !

*lm*(2)

*l*

_{aS}

*lb Y*

*m*

*l*(U , F)

### N

2*1 h*0

### N

### !

*lm*(2)

*l*

_{aS}

*l*b

### (

*Ylm*(U , F)

### )

*### N

2### n

,*where the amplitude M*0 depends on the differential cross-section for the reaction

*A( a , b) B, in the quantity wl* *we have introduced the variable m f J*B*2 J*C *instead of J*C

*and the relative amplitude h*0, given by

### (

see eqs. (11) and (14)### )

*h*0*f Nf*ba*(2m*0*, v*b) N2*/Nf*ba*(m*0*, v*b) N2,

(28)

is associated with the negative polarization of the decaying residue B.

*Equations (26), (27c) and (28) show that information on the polarization effects of*
the residual nucleus B induced by the first step of the sequential process
A( a , b) B( c *) C can be detected through the W-dependence of the differential*
multiplicity for the second step.

**4. – Semi-classical multiplicity**

In this section we shall derive a semi-classical expression for the b-c differential multiplicity which accounts for many of the observed features of the sequential emission of the highly as well lowly energetic particles from the fragments excited in a peripheral heavy-ion reaction.

Firstly, we consider a semi-classical picture that assumes a coordinate rotation to a more useful system

### (

see assumptions (21)-(23) and eq. (26)### )

, where the new**quantization axis is oriented in the direction of J**B which is at a certain angle L with

*respect to the z-axis*

### (

see eqs. (23)### )

and lies in a plane perpendicular to the reaction**plane and to the direction of a unit vector k**×0, close to the recoil direction of the

*decaying nucleus B [17], corresponding to an angle W*0*4 (p/2 1 j) with respect to the*

*x-axis.*

*The rotation of axes defined by the Euler angles (j , L , 0) and the coordinate*
system chosen in describing the decay B Kc1C

### (

see eqs. (21), (23), (25) and (26)### )

are shown in fig. 2.**Then the relative momentum k**c*of the pair ( cC ) has polar angles (w , W) and (U , F)*

*with respect to the space-fixed system and to the (k*×0*3 J*×B*, k*×0*, J*×B)-axes, respectively.

*Since the polar angles of J*B*-axis with respect to the (x , y , z)-axes are (L , p 1j)*

(see fig. 2), we have

*cos U 4cos L cos w2sin L sin w cos (W2j) ,*
*(29a)*

cot F 4

### (

*cos L sin w cos (W 2j)1sin L cos w*

### )

*/sin w sin (W 2j) .*

*(29b)*

**4**.*1. Equilibrium multiplicity. – Following the quantal treatment carried out in*
ref. [18], we assume the semi-classical replacement

### (

*see eq. (27b)*

### )

[18, 19]*wl(m) Aexp [2al*2*] exp [bm] ,*
(30)

*Fig. 2. – Illustration of the rotation defined by Euler angles (j , L , 0 ) and of the coordinate system*
used in describing the decay B Kc1C (see the text).

where
*a f (I 1MR*2_{) ˇ}2* _{/2 IT}*
C

*MR*2,

*(31a)*

*b f J*Bˇ2

*/IT*C

*(31b)*

*with M, R and I the reduced mass, the radius and the rigid-body moment of inertia of*
*the pair (cC), respectively, and T*C the nuclear temperature corresponding to the

*excitation energy e **Cin the nucleus C.

In a classical picture we can also utilize the relation [18]

### !

_{m}*exp [ bm]*

### (

*Ylm*(U , 0 )

### )

2*A lJ*0

*(ilb sin U) ,*

(32)

*where J*0is the zeroth-order Bessel function with an imaginary argument.

*Substituting eqs. (30)-(32) in eq. (27b), in the sharp cut-off approximation for the*
*coefficient Tl, converting the summation over l to an integral, since [20]*

*2 a*

0 Q

*l dlJ*0*(iql) exp [2al*2*] 4exp [q*2*/4 a] ,*

we obtain

### (

*see eq. (29a)*

### )

### (

*M(w , W , L)*

### )

E*4 C*E*exp [2g cos*2U] ,

(33)

*where C*E *is independent of w and W and the anisotropy coefficient g is given by*

*g f* *b*

2

*4 a* .
(34)

**4**.*2. Nonequilibrium multiplicity. – We associate the “direct” sequential decay*
B Kc1C described by aS b

### (

see eqs. (16)### )

with a situation in which the particles c are “promptly” emitted from peripheral regions of the nucleus B and we envisage that in the classical limit the particles c which escape from the rotating nucleus B get an additional velocity if they are emitted at the equator.So that for an estimate of the NE b-c multiplicity we can assume the emission of
**particles c in the equatorial plane with orbital angular momentum l parallel to J**B to

*dominate and, setting U 4p/2 (see fig. 2) and m4l in eq. (27c), write*
(2)*l _{Y}m*

*l* (U , F) A

### (

*Yll(p/2 , 0 )*

### )

*exp [il F] .*(35)

Besides, in line with the above arguments, we assume that the peripheral nature of the
*NE decay process is consistent with the hypothesis that only a “l-window” centered at*
*a certain l*0 contributes. So that for the energy-averaged element aS*l*b, in the
amplitude-phase representation

aS*lb 4h(l) exp [id(l) ] ,*
*we can write near l 4l*0

aS*lb Ah(l2l*0*) exp [i(l 2l*0*) x*0] ,

*where we have assumed the phase d(l) linear in l about l*0and

*x*0f

## g

ˇ*d(l)*

ˇ*l*

## h

*l*0

(37)

is the so-called quantal deflection function which describes somehow the “classical
trajectory” of the particles c and the nucleus C in their mean field characterized by the
*phase shift d [21].*

*From eqs. (35)-(37), for an estimate of the NE differential multiplicity (27c), we can*
write

### (

*see eq. (29b)*

### )

### (

*M(w , W , L)*

### )

NE*A NQ*(1)(F) N2*1 h*0*NQ*(2)(F) N2,

(38)

where we have defined the “single source” amplitude

*Q*(6)_{(F) f}

### !

*l* *h(l 2l*0*) exp [i(l 2l*0

*)(x*06 F) ]

(39)

and we have neglected the dependence of

### (

*Yl*

*l(p/2 , 0 )*

### )

*upon l since, for the*

*semi-classical case (l c 1 ), Yl*

*l(p/2 , 0 ) Al*1 O4*is a very smooth function of l.*

Recalling the peripheral nature of the direct NE decay process, if we express the
*amplitude h(l 2l*0) as a Gaussian distribution [22]

*h(l 2l*0*) Aexp [2(l2l*0)2*/4 l*2] ,

*from eq. (39), converting the summation over l to an integral, since [20]*

2Q 1Q

*dl exp [2l*2* _{/4 l}*2

*] exp [2itl] 42l(p)*1 O2* _{exp [2t}*2

*2*

_{l}_{] ,}

we finally obtain

### (

*M(w , W , L)*

### )

NE*4 C*NE*]exp [2l*2*(F 1x*0)2*] 1h*0*exp [2l*2*(F 2x*0)2]( ,

(40)

*where C*NE *represents all the inessential constants independent of w and W.*

In order to obtain the final expression of the semi-classical b-c differential
multiplicity, we shall assume that the spin orientation is governed by a distribution
*function L(L)*

### (

see eqs. (23) and fig. 2### )

, so that finally we have*M(w , W) 4*

### [

### (

*M(w , W)*

### )

E1

### (

*M(w , W)*

### )

NE_{]}

(41)

with

### (

see eqs. (29), (33) and (40)### )

### (

*M(w , W)*

### )

E 4*dL L(L)*

### (

*M(w , W , L)*

### )

E### N

*dL L(L) ,*(42)

### (

*M(w , W)*

### )

NE 4*dL L(L)*

### (

*M(w , W , L)*

### )

NE### N

*dL L(L) .*(43)

*For simplicity, in the following we shall assume for the function L(L) a Gaussian*
distribution

*L(L) 4exp [2(L2L*0)2/2 V2] .

**5. – In-plane differential multiplicity**

*The in-plane differential multiplicity corresponds to w 4p/2. In this case eqs. (29)*
become

*cos U 4sin L cos (W2j) ,*
*(45a)*

*cot F 4cos L cot (W2j) .*
*(45b)*

*It is seen from eqs. (40) and (45b) that when the dealignment is sufficiently small*
(L b 1), the NE in-plane b-c differential multiplicity is essentially given by a two
*component asymmetric (in general h*0c*1) pattern about the angle j 4W*0*2 p/2 (see*

*fig. 2), peaked at the angles W*1*4 j 2 x*0 *and W*2*4 j 1 x*0, respectively; moreover, if

*x*0*E j and h*0E 1, the b-c coincidence events appear with maximum probability on the

same side of the beam axis with respect to the direction of the “detected” projectile
residue. We note in passing that the values of the in-plane coincidence cross-section
*around W*1 *and W*2correspond to reaction process A( a , b) B with opposite polarization

of B, which, in a qualitative picture, may be somehow explained by the assumption that only one type of “semi-classical trajectory” predominantly contributes to the in-plane b-c angular correlation for either positive or negative angles with respect to the direction of the projectilelike nucleus b [21, 23].

*In the cases when L b 1 one can obtain an estimate of the angle j and the quantal*
*deflection x*0 by a simple inspection of the experimental in-plane angular correlation

*pattern around the “peak angles” W*1*and W*2, using the expressions

*2 j CW*2*1 W*1,

*(46a)*

*2 x*0*C W*2*2 W*1.

*(46b)*

Now it should be outlined that other models also predict double-peaked angular
*correlation as, e.g., the “hot-spot” model [5] where two maxima are predicted due to the*
process of “shadowing” of the light particles emitted by the target and reabsorbed by
the projectile (or vice versa).

Indeed here the deviation from left-right symmetry about a direction close to the one of the coincident projectile residue as well as the double forward-peaked shape in the angular correlation pattern does not necessarily imply that the light particles emerge from the contact zone between the two colliding nuclei (spatial localization). Actually, in a simple optical picture, we can interpret the sums appearing in eq. (39)

### (

see also eq. (40)### )

*as a beam of particles c emitted from a “l-window” centered about a*

*mean value l*0

*and extended over a narrow width Dl Al (l-localization).*

Note that, since the evaporative component is expected to dominate at backward
*in-plane angles, in practical use, the quantities C*E*, g, L*0and V can be obtained from

fitting, by the simple formula (42)

### (

*see eqs. (33), (42) and (45a)*

### )

, the in-plane and out-of-plane b-c angular correlation experimental data corresponding to backward in-plane angles.In conclusion of this section we utilize a semi-classical wave packet description [24]
*in order to estimate the mean time interval t*0between the formation of the nucleus B,

produced in the peripheral collision A( a , b) B, and the “fast” emission B KC1c.
In a simple classical picture we consider the composite system ( c1C ) rotating
*during the finite time t*0 *with relative angular momentum l*0 and with rotational

*now assume that the “deflection” angle x*0*is a sensitive function of NE decay time t*0,

*defined to be t*0**4 0 when the component of k**cin the reaction plane is in the direction of

**k**

×

0*, corresponding to the angle W*0*4 j 1 p/2 (see fig. 2), so that we get the linear*

relation

*2x*0*4 v*0*t*0.

(47)

*At the moment of formation of the decaying nucleus B, t*04 0, we represent the system

*( c 1C ) as a packet in the W-plane [24, 25]:*

*u(W , 0 ) A*

*dl exp [il(W 2W*0

*) 2 (l2l*0)2

*/4 l*2] ,

that is

*Nu(W , 0 ) N*2*A exp [2l*2*(W 2W*0) ] .

*After the time t*0 the wave packet evolves and expands so that, omitting a slowly

*varying function of t*0*, in the semi-classical case (l c 1) we obtain*

*Nu(W , t*0) N2*A exp [2l*2*(W 2W*0*2 v*0*t*0)2] .

(48)

*If we substitute eq. (44) into eq. (40) with h*04 0 and compare eqs. (40) and (45), we can

remark that the NE in-plane differential multiplicity is very similar to the angular
*probability Nu(W, t*0) N2. From the above rough picture we see that the assumption (44)

somehow idealizes the time dependence of the NE decay B Kc1C; for example the observed strongly forward-peaked in-plane angular correlation can be interpreted as an indication that the light particles c are emitted in decay times shorter than the rotational period of the nucleus B, corresponding to the time required for a hypothetical complete revolution of the composite system ( c 1C ).

**6. – Application**

In this section we consider the application of the semi-classical approach (41) to the
*C-a differential multiplicity for the sequential process*58_{Ni(}16_{O , C) Zn(a) Ni at 96 MeV}

laboratory energy [3, 4].

The experimental data shown in fig. 3 and fig. 4 are deduced from the ones of ref. [3]
*and ref. [4], respectively, and correspond to the a-particles, in coincidence with the*
C-ions detected at a laboratory angle of 2357 with respect to the incident beam,
emitted from intermediate nuclei Zn produced in the deeply inelastic collision 16

O 1

58

*Ni KC1Zn1Q assuming a window on the two-body Q-value [3, 4].*

The differential multiplicities of fig. 3 and fig. 4, given in units 1022 _{sr}21_{, are}

evaluated in the rest frame of the Zn with excitation energy corresponding to

*Q 42 35 MeV, that is the mean value of the above Q-window.*

*The in-plane differential multiplicity M(W) (see fig. 4) is plotted as a function of the*
*in-plane a-angle W, measured relative to the direction of the incident beam taken as the*

* x-axis (see fig. 1 and subsect. 3*.2).

*The out-of-plane differential multiplicity M(w) (see fig. 3) is plotted as a function of*
*the out-of-plane a-angle w, measured relative to an axis perpendicular to the reaction*
* plane, taken as the z-axis (see subsect. 3*.

_{2); so w407 corresponds to a direction }*perpen-dicular to the reaction plane, whereas w 4907 to an in-plane angle W421307 [3].*

*Fig. 3. – Best fit of the out-of-plane C-a differential multiplicity data, for the sequential process*
58_{Ni(}16* _{O , C) Zn(a) Ni at 96 MeV laboratory energy [3], with eq. (42) represented by the dashed}*
curve. The differential multiplicity, in units 1022

_{sr}21

_{, is plotted as a function of the out-of-plane}alpha-angle. See the text for explanations.

From an inspection of fig. 3 we consider the observation of the ratio between the
*yield in the reaction plane (w 4907) and the yield perpendicular to it (w407) as an*
**indication that the dealignment of the a-decaying nucleus Zn is small (see sect. 4 and***fig. 2). Thus, in order to evaluate the quantities j and x*0, we can utilize the approximate

*expressions (46). Their values so obtained are j 42 307 and x*04 2 45 7.

*Since the out-of-plane C-a coincidences are taken at backward angles [3], where the*
contribution from nonequilibrium emission is negligible, we can use these experimental
*data in order to fit the parameters (C*E*; g ; L*0; V) by the purely evaporative formula

*(42). Best x*2 _{values for the quantities (C}

E*; g ; L*0*; V), estimated within 5 %, are C*E4

*1.4; g 43.0; L*04 19 7; V 4 13 7. The corresponding best fits of the out-of-plane and

*in-plane equilibrium differential multiplicities, given by eqs. (29a), (33), (42) and (44),*
are represented by the dashed curves in fig. 3 and fig. 4, respectively.

*Finally, treating the quantities (C*NE*; l ; h*0) as adjustable parameters, the complete

experimental in-plane differential multiplicity [4] has been fitted by the formula (41),
*where the above-determined values of the quantities (C*E*; g; L*0*; V; j; x*0) were

*inserted. Best x*2 _{values for the quantities (C}

*Fig. 4. – Best fit of the in-plane C-a differential multiplicity data, for the sequential process*
58_{Ni(}16* _{O , C) Zn(a) Ni at 96 MeV laboratory energy [4], with eq. (41) represented by the solid}*
curve. The dashed curve is the best fit of the equilibrated part of the differential multiplicity with
eq. (42). The differential multiplicity, in units 1022

_{sr}21

_{, is plotted as a function of the in-plane}alpha-angle. See the text for explanations. The arrows indicate the directions of the projectile-like fragment (b) and target-projectile-like fragment (B) with respect to the incident beam in the laboratory

**system, namely of v**L

b**and v**LB, respectively (see fig. 1). In this case b f C and B f Zn.

*C*NE*4 2.1; l 4 2.3; h*04 0.3. The corresponding best fit of the in-plane differential

multi-plicity, given by eqs. (29), (33) and (40)-(44), is represented by the solid curve in fig. 4.
From eq. (28) we can define a positive alignment parameter on a quantization axis
**perpendicular to the reaction plane as (see subsect. 3**.2) (we omit the explicit indication
*of v*b)

*p*0*4 Nf*ba*(m*0) N2*/(Nf*ba*(m*0) N2*1 Nf*ba*(2m*0) N2*) 4 (11h*0)21.

*In our application we have, for Zn, p*04 0.77. This result may be explained by the

assumption [23] that, in the deeply inelastic collision58_{Ni(}16_{O , C) Zn at 6 MeV/nucleon,}

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