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 16O158Ni 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
O158Ni 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, JBpB) in the nucleus B to a narrow definite state
(e *C, JCpC) in the final nucleus C.
From now on, e *Xindicates the excitation energy of the state of definite spin JXand
parity pX in the nucleus X and mX the z-component of JX. The pair (xX) has relative
radial coordinate rx, momentum kx, velocity vx and energy ex. The spherical polar
angles (wb, Wb) of kb are defined in the ( A 1a ) centre-of-mass (c.m.) system, while kc
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, JBmB), associated with the unit flux in the partition ( aA ; ea), in
the asymptotic region is given by [8, 9] (v f (w , W))
Cb4 fba(e *B; mB, vb) fbfBOb.
(1)
Here the outgoing wave Obis given by
Ob4 (vb1 /2rb)21exp [i(kbrb2 hbln 2 kbrb1 s0b) ]
(2)
with hb and s0b the Coulomb field parameter and the S-wave Coulomb phase shift for
and B, respectively; the amplitude fba, related to the differential cross-section for the reaction A( a , b) B by means of ds dvb 4
!
m BNf ba(e *B; mB, vb) N2, (3)determines the population of the substate (e *B; JBmB), in the nucleus B, prepared by
detecting the particle b in the fixed direction of kb.
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; JBmB) to a given partition ( cC ; e *C, JCmC), in the
asymptotic region can be expressed by [10]
fB4 FBc(e *B, e *C; mB, mC, v) fcfCOc,
(4)
where the function Oc is expressed in a form analogous to eq. (2), fc and fCare the
normalized wave functions of the nuclei c and C, respectively, and the decay amplitude
FBcis defined by (5) FBc(e *B, e *C; mB, mC, v) 42 i
!
l (2) laJ ClmC, mB2 mCNJBmBb Q Q Sl(e *B, e *C); YlmB2 mC(v). In eq. (5) l is the relative orbital angular momentum of the pair (cC), Yml (v) are the spherical harmonics, aJClmC, mB2 mCNJBmBb 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 fB 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 fc and fC are normalized and the S
matrix is unitary [9], from eqs. (4) and (5) we have
!
mBmC
de *Bde *CdvdtNfBN24 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 ; ea) K ( bB; e*BJB) K ( bcC; e*CJCmC) we can substitute the wave function fB
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; mC, vb, v) ObOcfbfcfC, (7) where Fbc(e *B, e *C; mC, vb, v) 4
!
mB I fba(e *B, mB, vb) FBc(e *B, e *C; mB, mC, 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 vb and v, corresponding to the hatched angular sectors, subtended by the angle-fixed b-particle detector Db and angle-variable c-particle detector Dc, respectively. Labels L and cm refer to the laboratory and ( A 1a ) centre-of-mass system, respectively. The circle is centered at vL
Bwith the radius equal to the velocity vcof 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 jbcin the partition ( bcC ; e *CJCmC) through the surface elements of
area r2
bdvb and rc2dv, respectively, and defined by [8, 9]
jbc(mC; vb, v) 4vbrb2dvbvcrc2dv
dzNCbcN2,(9)
where the integration is taken over the totality of the internal variables.
If we recall the unit flux in the partition ( aA ; ea), jbcis just the double differential
cross-section d2
s for the sequential reaction ( aA; ea) K ( bB; e*BJB) K ( bcC; e*CJCmC).
So, by performing the integration in eq. (9), with the functions fb, fc and fC
normal-ized, the double differential cross-section for any final spin, from eqs. (7)-(9), is d2s dvbdv 4
!
m CN
!
mB Fba(e *B; mB, vb) FBc(e *B, e *C; mB, mC, v)N
2 , (10) where Fba(e *B; mB, vb) f I fba(e *B; mB, vb) . (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 2s dvbdv
N
ds dvb , (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 JB-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 *Band 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 e0
a f(e0)bD4
2Q 1Qde r(e , e0, 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 , e0, D), for the result to have physical sense.
Besides we assume that the amplitude fba (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 dvbp
C!
m BNf ba(mB, vb) N2 (14)and write
(
see eqs. (5) and (11))
o
d2s dvbdvp
C!
JCmCN
!
mB Fba(mB, vb)aFBc(mB, mC, v)bN
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 SNE 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 varyingfunction 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 SE, must be subjected
also to an e *C-energy averaging process
(
see eq. (13))
to yield a smoothly varyingfunction of energy e *C.
In line with the above arguments, the average of ( ds)Ein the interval D8 centered at
e *Ccan be approximately given in terms of just the quantity ( d2s)E, associated with the
mean value e *C, times the relative density of states of angular momentum JC 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
d2s dvbdvp
4g
d 2s dvbdvh
E 1g
d 2s dvbdvh
NE (17) withg
d2s dvbdvh
E 4!
mC!
lJC wl(JC)g
Tl Gh
N
!
mB pl(mB, mC; vb, v)N
2 , (18)g
d2s dvbdvh
NE 4!
mCN
!
lJC aSlb!
mB pl(mB, mC; vb, v)N
2 , (19) where (20) pl(mB, mC; vb, v) f (2)lFba(mB, vb)alJC, mB2 mC, mCNJBmBb YlmB2 mC(v) . In eq. (18) the quantity wl, related to the relative density of the available states (e *C, JCpC) in the nucleus C, describes the probability of orbital angular momentum ltransferred in the decay ( B ; JBe *B) K ( cC; lJCe *C) and we have assumed the
parametrization aNSEN2b 4Tl/G, where Tlis 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 ( d2s)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 kb3 ka(perpendicular to the reaction plane) and the x-axis along ka. Note that the sign convention is here chosen so that a positive in-plane angle W corresponds to an angle of kcwith respect to kain the reaction plane, such that ka3 kcD 0. So that we have WbE 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 Fba(mB, vb)
(
see eqs. (3) and (11))
forconvenience in the form
Fba(mB, vb) CNFba(mB, vb) Nexp [2ijmB] .
(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))
NFba(mB, vb) NCNFba(NmBN 4 m0, vb) N
(22)
with m0the component of JBalong the z-axis. In the processes here considered JBand
JCare usually large as compared to l so that we can replace the CG coefficients by their
asymptotic expressions [11]
aJClmC, mB2 mCNJBmBb 4dmml (L) (23a)
where dl
mmis the reduced rotation matrix with m f mB2 mC, m f JB2 JCand L given by
L 42cos21
g
mBJB
h
. (23b)
Then, if the above approximations and restrictions can be justified, from eqs. (21)-(23) we have (m f mB2 mC, m f JB2 JC)
(24) pl(mB, mC; vb, v) C (2)lNFba(NmBN 4 m0, vb) Nexp [2ijmC] ) 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 JCis in the same direction of l.
Then, by using the transformation [11]
Ylm(U , F) 4
!
mDl
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) )E4!
lm wl(m)g
Tl Gh
(
Y m l (U , 0 ))
2 , (27c)(
M(v))
NE 4 M0m
N
!
lm(2) laS lb Y m l (U , F)N
2 1 h0N
!
lm(2) laS lb(
Ylm(U , F))
*N
2n
, where the amplitude M0 depends on the differential cross-section for the reactionA( a , b) B, in the quantity wl we have introduced the variable m f JB2 JC instead of JC
and the relative amplitude h0, given by
(
see eqs. (11) and (14))
h0f Nfba(2m0, vb) N2/Nfba(m0, vb) 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 JB which is at a certain angle L withrespect 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 thedecaying nucleus B [17], corresponding to an angle W04 (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 kcof the pair ( cC ) has polar angles (w , W) and (U , F)
with respect to the space-fixed system and to the (k×03 J×B, k×0, J×B)-axes, respectively.
Since the polar angles of JB-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 [2al2] 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 1MR2) ˇ2/2 IT CMR2, (31a) b f JBˇ2/ITC (31b)
with M, R and I the reduced mass, the radius and the rigid-body moment of inertia of the pair (cC), respectively, and TC 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 ))
2A lJ0(ilb sin U) ,(32)
where J0is 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 dlJ0(iql) exp [2al2] 4exp [q2/4 a] ,
we obtain
(
see eq. (29a))
(
M(w , W , L))
E4 CEexp [2g cos2U] ,
(33)
where CE 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 JB to
dominate and, setting U 4p/2 (see fig. 2) and m4l in eq. (27c), write (2)lYm
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 l0 contributes. So that for the energy-averaged element aSlb, in the amplitude-phase representation
aSlb 4h(l) exp [id(l) ] , we can write near l 4l0
aSlb Ah(l2l0) exp [i(l 2l0) x0] ,
where we have assumed the phase d(l) linear in l about l0and
x0f
g
ˇd(l)
ˇl
h
l0(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))
NEA NQ(1)(F) N21 h0NQ(2)(F) N2,
(38)
where we have defined the “single source” amplitude
Q(6)(F) f
!
l h(l 2l0) exp [i(l 2l0
)(x06 F) ]
(39)
and we have neglected the dependence of
(
Yll(p/2 , 0 )
)
upon l since, for the semi-classical case (l c 1 ), Yll(p/2 , 0 ) Al1 O4is a very smooth function of l.
Recalling the peripheral nature of the direct NE decay process, if we express the amplitude h(l 2l0) as a Gaussian distribution [22]
h(l 2l0) Aexp [2(l2l0)2/4 l2] ,
from eq. (39), converting the summation over l to an integral, since [20]
2Q 1Q
dl exp [2l2/4 l2
] exp [2itl] 42l(p)1 O2exp [2t2l2] ,
we finally obtain
(
M(w , W , L))
NE4 CNE]exp [2l2(F 1x0)2] 1h0exp [2l2(F 2x0)2]( ,
(40)
where CNE 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 haveM(w , W) 4
[
(
M(w , W))
E1
(
M(w , W))
NE]
(41)
with
(
see eqs. (29), (33) and (40))
(
M(w , W))
E 4dL L(L)(
M(w , W , L))
EN
dL L(L) , (42)(
M(w , W))
NE 4dL L(L)(
M(w , W , L))
NEN
dL L(L) . (43)For simplicity, in the following we shall assume for the function L(L) a Gaussian distribution
L(L) 4exp [2(L2L0)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 h0c1) pattern about the angle j 4W02 p/2 (see
fig. 2), peaked at the angles W14 j 2 x0 and W24 j 1 x0, respectively; moreover, if
x0E j and h0E 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 W1 and W2correspond 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 x0 by a simple inspection of the experimental in-plane angular correlation
pattern around the “peak angles” W1and W2, using the expressions
2 j CW21 W1,
(46a)
2 x0C W22 W1.
(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 l0and 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 CE, g, L0and 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 t0between 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 t0 with relative angular momentum l0 and with rotational
now assume that the “deflection” angle x0is a sensitive function of NE decay time t0,
defined to be t04 0 when the component of kcin the reaction plane is in the direction of
k
×
0, corresponding to the angle W04 j 1 p/2 (see fig. 2), so that we get the linear
relation
2x04 v0t0.
(47)
At the moment of formation of the decaying nucleus B, t04 0, we represent the system
( c 1C ) as a packet in the W-plane [24, 25]:
u(W , 0 ) A
dl exp [il(W 2W0) 2 (l2l0)2/4 l2] ,that is
Nu(W , 0 ) N2A exp [2l2(W 2W0) ] .
After the time t0 the wave packet evolves and expands so that, omitting a slowly
varying function of t0, in the semi-classical case (l c 1) we obtain
Nu(W , t0) N2A exp [2l2(W 2W02 v0t0)2] .
(48)
If we substitute eq. (44) into eq. (40) with h04 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, t0) 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 process58Ni(16O , 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 sr21, 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 58Ni(16O , C) Zn(a) Ni at 96 MeV laboratory energy [3], with eq. (42) represented by the dashed curve. The differential multiplicity, in units 1022sr21, 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 x0, we can utilize the approximate
expressions (46). Their values so obtained are j 42 307 and x04 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 (CE; g ; L0; V) by the purely evaporative formula
(42). Best x2 values for the quantities (C
E; g ; L0; V), estimated within 5 %, are CE4
1.4; g 43.0; L04 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 (CNE; l ; h0) 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 (CE; g; L0; V; j; x0) were
inserted. Best x2 values for the quantities (C
Fig. 4. – Best fit of the in-plane C-a differential multiplicity data, for the sequential process 58Ni(16O , 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 sr21, 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 vL
band vLB, respectively (see fig. 1). In this case b f C and B f Zn.
CNE4 2.1; l 4 2.3; h04 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 vb)
p04 Nfba(m0) N2/(Nfba(m0) N21 Nfba(2m0) N2) 4 (11h0)21.
In our application we have, for Zn, p04 0.77. This result may be explained by the
assumption [23] that, in the deeply inelastic collision58Ni(16O , C) Zn at 6 MeV/nucleon,
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