Quantum-mechanical treatment of diabatic neutron emission in heavy-ion collisions

Nuclear Physics A438 ( 1985) 253-280 @ North-Holland Publishing Company

QUANTUM-MECHANICS TREATMENT OF DIABATIC NEUTRON EMISSION IN HEAVY-ION COLLISIONS WOLFGANG

GASSING

GSI Darmstadt, D-6100 Darmstadt, F.R. Germany Received 31 August 1984 Abstract: The diabatic precompound emission of neutrons is studied in a finite time-dependent twocenter shell model for central cothsions of *‘Zr f ‘“Zr and 96Zr + *6Zr. For laboratory energies between 8 MeV/u and 20 MeV/u the neutron multiplicity is found to be significantly enhanced for 96Zr+96Zr as compared to 9CZr+9“Zr. The angular distribution of the emitted neutrons is forward-backward peaked in the center-of-mass system, while the energy spectra show an exponential tail. with a slope parameter To roughly independent of the scattering angle. A close relation between the diabatic emission process, the Fermi-jet mechanism and the exiton model is established.

1. Intr~uctjon Collisions between nuclei at energies well above the Coulomb barrier are accompanied by a considerable yield of emitted light particles 1- ‘*), which apart from the decay of the equilibrated heavy fragments show a nonequilibrium component. The latter is expected to originate from the early stage of a heavy-ion motion is still far from equilibrium. reaction, where the single-particle Consequently, precompound or nonequilibrium light-particle spectra should reflect the coupling of the relative motion to the intrinsic degrees of freedom and the related approach to equilibrium. This interesting perspective has motivated several theoretical studies 19-3s ), which roughly group into four categories: the exciton model 13-23), the Fermi-jet approach 24-28S33), the hot-spot model 31,32) and the diabatic approach 34*3s). In addition, many experimentalists prefer to parametrize their data by “moving hot sources” *‘, 18). While the first three types of concepts have been worked out to a rather high degree of sophistication and been applied with varying success to experimental data, the diabatic approach was presented only in its semiclassical version 34,3s). A proper quantum-mechanical formulation is presented in sect. 2 with particular emphasis on appropriate time-dependent s.p. states in the continuum. The characteristic properties of diabatic neutron emission are investigated in sect. 3 with respect to shell effects in the neutron multiplicity for 96Zr+96Zr and 90Zr+90Zr. In addition, the angular and energy distributions of the emitted neutrons are evaluated for central collisions of 96Zr+962r at 12 MeV/u and 20 253

W. Cussing 1 Diahutic

254

neutron emission

MeV/u, respectively. In sect. 4 the diabatic with the Fermi-jet approach and the exciton is seen

to form

shortcomings improvements

a link

between

of the present are proposed.

2. Single-particle

concept and its results are compared model. Diabatic single-particle motion

the different diabatic

concepts.

approach

dynamics and collective

are

Finally,

in sect.

discussed

and

5 the

possible

nuclear motion

Nucleus-nucleus collisions up to roughly 20 MeV/u bombarding energy are expected to be well described by time-dependent Hartree-Fock (TDHF) 36. 37) during the initial phase of the reaction because nucleon-nucleon collisions appear to be Pauli-blocked considerably 38-40). Within this picture the single-particle (s.p.) motion is completely determined by the corresponding mean-field U(po(r)) which depends on time via the one-body density pa(t). For central collisions, axial symmetry around the beam axis holds, such that zl(p,z; t) is a function of the cylindrical coordinates p and z only. For illustration, the neutron mean field L’(p, z; t) is shown in fig. I for p = 0.25 fm in terms of contour lines corresponding to - 2, - 10, -30, -60 and -70 MeV, respectively. The system studied within TDHF is shKr+ 168Er at 8.28 MeV/u for zero impact parameter, which was analyzed with respect to s.p. levels earlier4’ ). One observes two effects: (i) the U (0=025fm,z,tl

z [fml Fig. 1. The TDHF potential L’@ = 0.25 fm,z; I) for neutrons in the case of a central collision of s6Kr + 16sEr at 8.2 MeV/u. The contour lines marked 1 to 5 correspond to - 2, - 10, - 30, -60 and - 70 MeV, respectively.

W. Cassiny 1 Diahuric neutron emission

boundaries

(cuts) of the mean field show a rather

smooth

255

time dependence,

and (ii)

the diffuseness (difference in z between the cuts) is roughly constant in time. Similar conclusions may be drawn from the same type of plot for 40Ca +40Ca at 20 MeV/u from ref.42). This suggests approximating the self-consistent mean field U@,(t)) by a sp. potential V(p, 2; q(t)). which depends explicitly on a few collective parameters q(t). Indeed, it could be shown in ref. 41 ) that during the initial phase of the heavy-ion reaction the TDHF dynamics are rather well reproduced within a two-center oscillator model even by restricting to a single collective parameter, the relative distance R(r) between the two centers of masses. Consequently, s.p. motion may be described by solutions of the s.p. Schrodinger equation {( -h2/2M)A+

V(x, q(t))-ihC/c:t}lt,hl(x,

t)) = 0,

(2.1)

where V(x. q(t)) denotes a suitable time-dependent s.p. potential, x denotes spin, isospin and position of the nucleon and q(t) a set of time-dependent collective parameters. The evaluation of the time-dependent s.p. states IC/,(t) from eq. (2.1) is performed along the following line. We start by approximating Ii/,(t) in terms of travelling diabatic s.p. states @,Jq, 4) from a conventional two-center shell model, which yields collective mass parameters B,,(q) and a dynamical nucleus-nucleus potential via s.p. matrix elements of Z/dq, and the two-center hamiltonian H,(q). The latter dynamical quantities determine the time dependence of the collective parameters q(t) via a coupled set of second-order differential equations even in the case of time-dependent s.p. occupation numbers n,(t) (subsect. 2.1). Two-body collisions, furthermore, are treated in a relaxation ansatz for n,(t) and their effect on the collective path q(t) as well as on the s.p. potential V(x,q(t)) is included explicitly. For given V(x, q(t)), a subset of sp. states $,(t) close to or above the Fermi level is evaluated by direct integration of the Schrodinger equation (2.1) for a corresponding finite two-center shell model in order to obtain appropriate s.p. states in the continuum (subsect. 2.2). The decay of the s.p. states due to two-body collisions is described finally in subsect. 2.3.

2.1. THE DIABATIC

LIMIT

The solution of eq. (2.1) for all s.p. wave functions 4,(t) is of the same complexity as the full TDHF solution, since the time dependence of the collective parameters is not known a priori. The Schrodinger-fluid picture43) helps in this respect as it suggests to separate a collective phase and a s.p. phase from Ii/,(t) [refs. 43. “)I, i.e. ‘df’e&‘) - M wyx, q, 4)

+s(q)

(2.2)

256

W. Gassing / Diabatic neutron emission

with orthonormal stationary states 4=(q) and the initial condition Cap(O) = a,,. The quantities sP denote the diagonal elements of the s.p. hamiltonian in the states 4B,

Assuming the collective velocity potential to be linear in 4, i.e.

W(x947 4) =

cn 4,w,(q),

(2.4)

the time dependence of the expansion coefficients C,,(t) becomes very smooth (C,,(t) z 6,,) if the states 4B fulfill the diabaticity criterion45)

{m.-vw; v-W,}l&(q))

= 0

(2.5)

c!# P.

(2.6)

as well as the secondary condition45)

Ha, = oAIHo(q)l4p)= 02

The construction of stationary diabatic states $Jq), which obey (2.5) and (2.6), has been studied in detail in ref.45) within the methods of maximum overlap and maximum symmetry, respectively. For clarification, the s.p. hamiltonian H,(x, q) in eq. (2.3) is assumed to be given by the axial-symmetric two-center oscillator model 45,46)

&k

4) = T + v,(p,z)+ v,.,.+ v, T = ( -h2/2M)A,

UP, z) =

+Mo;,p2 ++Mo:,(z-z,)~ $Mu&p2 ++Mw;Jz-zJ~

(2.7)

(2.8)

for z < 0 for z > 0,

(2.9)

where the centers of the oscillators are given by z1 and z2. The spin-orbit potential is approximated by v,,,, = -Ichw(21~s+jd2}

(2.10)

is defined via the model parameters with K and p from ref. 47), while ClT, o2 = i{o$, +o;2+w:l +cL&).

(2.11)

W. Gassing 1 Diabaric neutron emission

Finally

the neck potential

V, is determined

257

by45)

v; = i!M (d,of,(z- Zij2+ .qi(Il;,y2}e(z; -

lzl)(z - z;)J

(2.12)

for z; -C z < z;, where the coefftcients di and .qi as well as z’, and z; are ftxed by matching the potential and its first derivatives at z’,, and z;. This procedure leaves freedom for a neck parameter r which is illustrated in fig. 2.

I

vo

I

t----AZ---+

model with model parameters.

The neck

degree of freedom is defmed by 5 = a/b while the dashed line smoothly joints the separated

Fig. 2. The mean lield V@ = 0. I) of the two-center

oscillators

(i.e. continuous

Adopting parameters

up to the first derivative).

oscillator

The depth of the mean lield is denoted by V,.

the condition of volume conservation 45.46), four transparent are finally left. These are the distance between the two

collective oscillator

potentials AZ = z2 -z,, the deformation 6 = w,,/ol,, = q/~~,, the neck size the mass asymmetry a. For completeness and transparency, the construction of the diabatic states along the line of the maximum symmetry method4’) is reviewed, since a moditication is introduced in order to account for the correct asymptotic structure of the nuclei. The construction starts by evaluating the eigenstates of the hamiltonian 4s) H&x, 4) = 7-+ v,(P, z),

5 and c$Jq) slight shell @(q)

(2.13)

which are smooth functions of the collective parameters q and define the corresponding velocity fields QnVwn(q) via (2.5). Asymptotically appropriate shell model states 4,(q(m)) are obtained by diagonalizing H,(Az --, q), (2.7), in the basis c#J~.This yields (2.14)

258

W. Cussing / Diabatic neutron emission

Stationary diabatic states 4a(q), which fulfill the diabaticity as the secondary condition (2.6), are then given by

criterion

da) = ;&l@(q) and travelling

diabatic

states

(2.5) as well

(2.15)

@Jq,tj) by

@&4) = exp (+WMPwAq) i

n

I

4.(q).

(2.16)

The latter states have been shown to yield reasonable solutions to the Schrodinger problem (2.1) at least for the collective degree of relative motion in central collisions of symmetric systems for collective energies between 1 MeV/u and 12 MeV/u above the Coulomb barrier 48). Denoting by n, the occupation number of the travelling the total energy of the sytem is given by 4534g)

E=

diabatic

states @Jq,4),

c n,<@,W,I@,) L1

=

Tcoll + I/,(q)

(2.17)

with (2.18)

I/,(q) =

c h%(q).

(2.19)

01

The collective

mass parameter

B,,

in (2.18) reads45)

Bnm= M c dd4Wwn). WY,JI~,)

(2.20)

11

and is identical to the corresponding irrotational flow value. One has to be careful in evaluating the potential term VP(q) according to (2.19) because Coulomb interactions are not properly described in the two-center hamiltonian (2.7). This deficiency, however, can be corrected by subtracting the term T/,(q) = Cs,(4)4(q), 01

(2.21)

where n:(q) denote the occupation numbers of the s.p. states in the configuration of lowest total energy for fixed q, and replacing &(q) by an adiabatic potential K,,(q) [ref.4g)]. The latter quantity e.g. may be calculated in the liquid-drop

W. Gassing / Diabatic neutron emission

model 50) for fixed collective coordinates effective nucleus-nucleus potential

q. This amounts

259

to considering

the

(2.22) which together with To,, (2.18) determines the time dependence degrees of freedom by total energy conservation49),

of the collective

(2.23) with B,, = B,, according to (2.20). The coupled set of second-order differential equations (2.23) is uniquely determined by stationary quantities, i.e. the diabatic s.p. levels E,(q) (2.3) and the diabatic states c&,(q) (2.15). Alternatively to eq. (2.20) the collective mass parameters B,, may be evaluated by the familiar cranking expression 45). It should be stressed, that B,,,,,(q) as well as I/,(q) are given by eqs. (2.20) and (2.22) even in case of time-dependent s.p. occupation numbers n,(t). Summarizing this section about the diabatic limit of eq. (2.1) we have obtained a unique way to determine the time dependence of the collective degrees of freedom introduced in (2.1) from stationary quantities. This allows one to study s.p. states $,(t) independent of the rest of the occupied states +@(t) (p # a), an advantage with respect to TDHF in particular for heavier systems. Furthermore, an appropriate s.p. she11 structure is obtained without major difficulties via eqs. (2.14) and (2.15). 2.2. A TIME-DEPENDENT

FINITE

TWO-CENTER

SHELL

MODEL

The mode1 hamiltonian (2.7) could be shown to reproduce the collective aspects of TDHF dynamics in the case of central collisions of heavy nuclei from 8 MeV/u to 20 MeV/u [ref.41)]. Since the time dependence of the collective degrees of freedom is basically determined by bulk properties (2.20) and (2.22), i.e. by large sums over s.p. matrix elements, slight changes in a few s.p. wave functions show almost no effect on the collective path. The travelling diabatic states Qm(q,4) (2.16) appear to be well suited for determining q(t). On the other hand, they are no longer appropriate for high-lying states if particle emission can occur. This defect arises from the infinite walls of the two-center mode1 (2.7) and the question comes up how to modify the mean field properly. The results of self-consistent calculations again serve as a guide. Restricting the discussion to neutrons, the TDHF mean field U(po(t)), of course, is finite (fig. 1) and its central depth is constant throughout time except for tiny local changes in the order of l-3 MeV. This statement holds for laboratory energies of 8 MeV/u (fig. 1) as well as for 20 MeV/u [ref.42)] and suggests introducing a simple cut-off

260

parameter

W. Cussing / Diabatic neutron emission

V, in the infinite s.p. potential (2.7) for neutrons, i.e. V(x, e) =

V,(P,Z)+K.,.+~~-~~ 0,

for V,+V,.,.+l$ otherwise.

G V,

(2.24)

For illustration, this mean field is displayed in fig. 2 for p = 0 and V,.,. = 0. The depth of the s.p. potential V, is uniquely determined by the asymptotic Fermi level .sr(co) and the experimental neutron binding energy .sB[ref. “‘)I,

v, = .+(Co)+Eg.

(2.25)

For the systems studied in sect. 3, we obtain +(co) = 43.0 MeV from (2.7) and sr, = 11.9 MeV [ref. “)I for 90Zr+90Zr, +(co) = 46.7 MeV and sr, = 7.9 MeV [ref. !’ )] for 96Zr + 96Zr. In spite of the completely different shell structure of the systems, the depth of the s.p. potential V, turns out to be almost the same, i.e. 54.9 MeV and 54.6 MeV, respectively. The latter property also holds for static HartreeFock calculations in case of neighbouring isotopes s2). Due to the simple manipulation (2.24) all low-lying states in the Schriidinger problem H&cl)

= T+ l+,q)

(2.26)

are almost identical with the solutions of H,(x, q) (2.7). Only high-lying states close to the Fermi level and essentially those in the continuum are affected. This implies that the time-dependent solutions of (2:l) again are given by travelling diabatic states (2.16) for stationary s.p. states well below the Fermi level, which predominantly determine the collective equations of motion. For the neutronemission problem, however, states which are shifted high up above the local Fermi level and couple to the continuum 34*35), are completely missed by the approximation (2.16) and have to be evaluated directly from the Schrodinger equation (2.1). The numerical integration of eq. (2.1) with V(x, q) given by (2.24) is still a very time-consuming project, since it requires solutions for the full three-dimensional problem. Within the spirit of diabatic s.p. motion34*3s*45*49* 53), however, we may write the solution q,(t) as a linear combination of states with higher symmetry in analogy to eq. (2.15), i.e. (2.27) where the expansion coefficients das are determined in the same way as in (2.14) and (2.15) by diagonalizing the full hamiltonian (2.26) in the basis states 4j(q(m)) from (2.13). The underlying argument is, that the nodal structure of the wave

W. Gassing / Diabatic neutron emission

function +,(t) remains couphng L. in (2.26) The latter assumption by a constant number

261

unchanged throughout the reaction or that the spin-orbit does not introduce dynamical changes in the wave function. implies that the potential term V,,,, (2.10) may be replaced in the Schriidinger problem (2.1) leading to

{ T + V&p,z) + Vs- V. - V.i“’ - ih~/i3t}cl@(f)) = 0

(2.28)

with the initial condition f#$(t = 0) = exp[( +iM/h)dw,(q(t

= O))]@(q(t

= 0)),

(2.29)

while the index c denotes the finite cutoff of the s.p. potential according to (2.24). In eq. (2.29) d is given by the asymptotic relative velocity of the colliding ions and wo by

woo, z; q)(t = 0)) = where A, and A, determined by

-f&!/(4 +A,k, -4A4

are the corresponding

+ 4)lzL

for z > 0 for z c 0,

masses. The quantity

(2.30) Vii”. is finally

with 4, from (2.14). The reduced Schrodinger problem (2.28) is quite easily solvable within the code ADIP [ref. ““)I which requires the special axial symmetry of (2.28). Apart from the fact that the time integration in (2.28) may be carried out independently for each wave function and shell effects can be included without major difficulties, the Schriidinger problem (2.28) has the additional advantage of allowing for the definition of the S-matrix. The latter property is due to the linearity of the problem in contrast to TDHF [ref. ““)I. On the other hand, fragmentation or condensation phenomena of nuclear matter, which appear to be caused by nonlinearities involved in the TDHF equations 56), cannot be described. 2.3. APPROXIMATE

TREATMENT

OF TWO-BODY

COLLISIONS

A lot of work has been devoted in recent years to extensions of TDHF including the effects of two-body collisions from Boltzmann-type collision terms 57). Expansions of the one-body density matrix in terms of time-dependent s.p. wave functions q,(t), i.e.

ptr,r’ ; t) = C ~.&M,tr Mb

; Ws*tr’; 0,

(2.32)

262

W. Gassing / Diabatic neutron emission

yield master equations for the occupation probabilities n,(t) = p,,(t) [ref. ‘“)J, which in lowest order may be approximated by the relaxation ansatz35’49’ 53) f% =

The equilibrium Fermi functions,

occupation

(- Vxx)[~,(t)- ri,(clY I4 7-11. probabilities

(2.33)

T) of eq. (2.33) are given by

ii,(q,p,

fi,(q,i4 77 = (1+expI(&,(q)--CLq))lT(q)))-‘,

(2.34)

where the chemical potential p and the temperature T are determined as functions of q by the total number of nucleons and the total excitation energy, respectively. The local equilibration time rTocin eq. (2.33) is assumed to be given by 49, 53) floe = 3X/v,.

(2.35)

In eq. (2.35) vr denotes the Fermi velocity and Ian paths 3s) ;1, =

average value of the mean free

C{(&,-p)2$7c27-2)-’

(2.36)

with C z 700 MeV2. fm [ref. ““)I in rough agreement with computations by Collins and Griffin 39). The time dependence of the s.p. occupation numbers n,(t) induces a change in the effective equation of motion (2.23) for the collective degrees of freedom. With respect to the mass parameters B,, (2.20) this explicit time dependence may be neglected because these quantities depend only on the s.p. density pO, i.e.

4). Vw,(r, q)pdr, q) B,,(q) = M d3rVw,(r,

(2.37)

with

fdr, 4) = C Q#b(r,4)12.

(2.38)

a

The driving force F,(q, t) = - iYV”(q)/8qn, however, becomes non-local in time 49, 53). This may be verified easily by inserting the formal solution of eq. (2.33) for n,(t) in eq. (2.23), which reads in the harmonic limit

+

CC

OLm s

’dt’&&‘)-d&,-p aqrn

aE,-p at-i

-

A

a4.

m-k4

exp

W. Cussing

The

set of eqs.

dissipative

diabatic

(2.39) are dynamics

: Diaharic

the effective (DDD)4y.

neutron

emission

equations s3), which

of motion uniquely

263

in the determine

theory

of

the time

dependence of the mean field V(x, q(t)). While the approach to local equilibrium requires roughly three subsequent twobody collisions (cf. eq. (2.35)). the mean lifetime r, of a highly excited state is given by the average time for a single nucleon-nucleon collision, rj, = (C/I.,)((E;.--)~+IC’T~)-’

(2.40)

which should with C from (2.36). This implies a “never come back” hypothesis, approximately be fulfilled because the equilibrium occupation numbers ti? are close to zero for high-lying states with ay - p 2 8 MeV. In the following these particular states, which are relevant for diabatic s.p. emission (cf. fig. 3). are denoted by the with eq. (2.40) the time dependence of the corresponding index 7. In accordance occupation numbers should follow from iI,(t) =

-r., ‘n,,(t).

(2.41)

Since the high-lying states IC/$t) are not treated in the travelling diabatic limit (2.16) but are evaluated from eqs. (2.27) and (2.28), a slight inconsistency arises in the calculation of the lifetime tY because the Fermi level p(q(t), T) (corrected by the potential depth V,) is evaluated within the travelling diabatic basis @,(q,d) of the infinite two-center problem (2.7). while s.,(t) is given by (2.42)

with d,, from eq. (2.27). The uncertainty of the coefficient C in (2.40), however, is of the order of +30’;/, [ref. ‘“)I such that the latter mismatch can be disregarded. In summary, the theory of dissipative diabatic dynamics 49*53) has been extended to account properly for time-dependent s.p. states $,(t), which are shifted in the continuum via the direct coupling to the collective degrees of freedom. The time dependence of the collective coordinates q(t) is entirely determined by the set of second-order differential equations (2.39), in which collective mass parameters are given in terms of the irrotational flow quantities (2.37) and the time-dependent forces by derivatives of stationary diabatic s.p. levels. Furthermore, the decay of the excited states due to two-body collisions is described simultaneously via eqs. (2.40) to (2.42).

264

W. Carsiny 1 Diabatic neutron emission

3. Diabatic neutron emission in central nucleus-nucleus collisions

Precompound neutron multiplicities resulting from the diabatic shift of s.p. levels have been calculated in a semiclassical framework for central collisions of “6Kr + *68Er and 92Mo +92Mo up to bombarding energies of 20 MeV/u [refs. “*“)]. It was found that the highest multiplicities are obtained from collisions of equal-mass nuclei. Furthermore, these multiplicities are expected to depend sensitively on the shell structure of the colliding ions3’). In order to investigate the latter question, the systems 90Zr +90Zr and 96Zr +96Zr appear as promising candidates since there is a gap of a few MeV between the N = SO and N = 56 neutron shell. First of all, the initial conditions of two-center shell model parameters in (2.7) have to be fixed. Assuming the relative distance between the two spherical nuclei to be 17 fm, .we have wp, = op, = Ok, = c+ 5 = 1 and dz = zz-zr = 17 fm. At bombarding energies a few MeV/u above the Coulomb barrier the dynamical role of the neck parameter r turns out to be completely negligible in the initial phase of the reaction. The dynamical role of the other collective parameters, i.e. the distance AZ between the centers of the oscillators and the deformation 6 = oJo,, furthermore, may be well described by a single collective degree of freedom4s), the relative distance R defined by

R(Az.6) = 2 Zpo(r; AZ, S)d3r/ s

p,,(r; Az, 6)d3r,

(3.1)

s

where the integration is confined to positive z-values and po(r; AZ, 6) denotes the nuclear density distribution. Consequently we may assume 6 = 1 and are left with a single. collective parameter R(Az). The eqs. (2.13) to (2.15) then determine the diabatic states 4,(R) as well as the corresponding diabatic levels E,(R) (2.3) which are shown in fig. 3 for 90Zr+90Zr and 96Zr+96Zr together with the corresponding local Fermi level +(R) (full dots). Since we are interested in diabatic neutron emission, only levels initially occupied at R =: 17 fm for E,(R) 2 36 MeV are displayed. In case of 96Zr+96Zr there are two additional levels crossing +(R) as compared to 90Zr +90Zr, which are expected to contribute significantly to the diabatic neutron-emission process. For reasons of notation (cf. sect. 2) all levels which cross t+(R) carry the index 7 in the following. It should be noted additionally, that each level in fig. 3 corresponds to two states 4. because of time reflection symmetry. The evaluation of the collective mass parameter BRR(R) is described extensively in ref.45) and the determination of the collective path R(t) in ref.35). With the knowledge of R(t) the time-dependent wave functions $,(t) can be uniquely calculated from eqs. (2.27) to (2.29). The initial collective velocity d in (2.29) is given by the center-of-mass energy I?,.,,. and the value of the Bass potential 59) at

n

38 -

38 -

40 -

+ 56

i

z z

w d

g

38 -

40-

42-

44-

W 4gd E 43-

W E 54

5

‘%R

+

‘%R

Fig. 3. Diabatic neutron levels for ‘Qr + 90Zr and 96Zr +96Zr evaluated with the hamiltonian (2.7) as functions of the relative distance R between the nuclei. For 96Zr+96Zr we observe two additional levels which cross the Fermi level (solid dots) already for R > 13 fm. With increasing overlap these levels are shifted into the continuum and should enhance the neutron multiplicity significantly when compared with collisions of “Zr + 90Zr.

P E

I 42W

‘%R

56

NEUTRONS

E

266 R =

W. Cussing 1 Diabatic neutron e~iss~an

17 fm, which was used for I/,,(R) in (2.22), i.e.

3.f. SHELL EFFECTS IN THE PRECOMPOUND ISOTOPES

NEUTRON

MULTIPLICITY FOR Zr

The integration of the Schriidinger equation (2.28) is performed within cylindrical coordinates on a grid with 51 x 101 mesh points in p and z together with the simple first-order differential eq. (2.41) for the occupation probabilities n,(t). Since for t + 00 n?(t) goes to zero according to (2.41), a cut-off time t, has to be introduced. for ny(t) because the emission of particles into the continuum is expected to be no longer reduced by two-body coilisions. Simple inspection of the time-dependent s.p. density shows, that the average time a particle (wave function) needs to be emitted into the continuum is given by At = t,- t, = 4r,At/(u,+k,)

(3.3)

with r0 z 1.2 fm. In eq. (3.3) ur denotes the Fermi velocity and AC the relative velocity at the contact distance R, = Zr,Af,

(3.4)

while t, is defined by the contact time R(t,) = R,.

(3.5)

The simple formula (3.3) corresponds to the average time a nucleon in nucleus A, with velocities close to +Z+M (with respect to the moving wall of A,) needs to transverse nucleus A, up to the boundary of A,. Indeed, wave functions localized initially in nucleus A, contribute to the continuum predominantly (> 90 %) at the side of nucleus A, (cf. sect. 4). This is clearly a quantum analogue to the Fermi-jet idea, that a nucleon travels through the other nucleus without being reflected at its moving wall 2’ ). For illustration, the continuum density P&l

2;

$1=

cY JqwqP~

2;

a2

(3.6)

is displayed in fig. 4 at t = 6.5 x 10mz2 s for a central collision of 96Zr + 96Zr at 20 MeV/u in terms of a raster plot. In order to show the contribution, which is directly emitted into the continuum, the density pC is suppressed for T = @t+Z2)* < rc = 12 fm. The continuum density is clearly seen to be forward-

261

W. Casing / Diabaric neutron emission

96 ZR + 96 ZR (20 MEV/U)

t=65 BS

Fig. 4. Raster plot of the continuum density p,(p, z; t) (3.6) at t = 6.5 x 10e2’ s in cylindrical coordinates for a central collision of 96Zr+96Zr at 20 MeV/u. The beam direction is identical with the z-axis. In order to show the neutron distribution emitted into the continuum, the density of the residual compound-like system for (p’ + z ’ )“’ < 12 fm has been suppressed.

backward coordinate

peaked with respect to the beam (z) direction. space yields the neutron multiplicity at time t, M,(t)

= 271 pp&,z;

Integration

t)0(z2 +p2 -r;)dpdz.

over

(3.7)

The observed neutron multiplicity results from the saturation value of M,(t) at large times. Numerical results for the neutron multiplicity M, (3.7) are shown in fig. 5 for 90Zr+90Zr (symbol W) and for 96Zr +96Zr (symbol 0) at laboratory energies from 8 MeV/u

to 20 MeV/u.

As an additional

s

z;t M: = 27~ PCW,(P> Y

information

the quantity

+ oo)128(z2+P2-r:)dpdz

(3.8)

is indicated in fig. 5 for the 90Zr system (symbol 0) and 96Zr + 96Zr (symbol O), which is the neutron multiplicity in the absence of two-body collisions. Both quantities show a significant enhancement of the precompound neutron yield for 96Zr+96Zr as compared to 90Zr + 90Zr. For comparison, the precompound neutron multiplicity obtained for 92Mo + 92Mo in the semiclassical approach 35) is displayed as well (symbol *). These values have to be compared with the quantum-mechanical results (3.7) for 90Zr +90Zr (symbol n) as it involves the same number of neutrons. The encountered deviations in the multiplicity up to a factor 2 indicate that the semiclassical approach may serve only as a rough guide for diabatic emission processes. The thresholds for the precompound emission are numerically rather uncertain

W. Cussing ! Diubaric neutron emission

268

4-

D0 0

0

3-

9% +g6zr *

)t

- 8-

- - - -9-m;---r

*I

0.’ 1’

3t ‘

I 10

1

L

I

I

t 15

t

I

t

‘

I 20

E lab /A [MeVI Fig.

5. Neutron

(symbol

l)

multiplicities

as a function

in the absence of two-body 96Zr + 9“Zr,

respectively.

(3.7) for central

of the bombarding coliisions For

are indicated

comparison,

approach

collisions

of 9oZr +90Zr

(symbol

a)

energy per nucleon. The precompound neutron

‘r) for 92Mo+92Mo

and 96Zr + ‘OZr multiplicities

(3.8)

by open squares and circles for 90Zr+PoZr

and

multipijcities

evaluated

in

the

semiclassical

are displayed by *.

since a very high numerical precision, i.e. small mesh sizes (< 0.2 fm) and time steps, is necessary. However, it is found empirically that the multiplicities increase almost linearly with the relative velocity d above the Coulomb barrier, This relation is displayed in fig. 6 where the multiplicities M, and M,Oare plotted versus ((E,.,.- V,)/A, jf, The notation is the same as in fig. 5. Approximating the “data” by straight lines. the cuts with M, = 0 determine the production thresholds to be at 6.5 MeV/u for 90Zr+90Zr and 5.9 MeV/u for 96Zr+96Zr. In accordance with the higher multiplicities for the g6Zr system, we also find a lower threshold which directly reflects the intuitive picture of shell structure effects from the diabatic twocenter level schemes (fig. 3). 3.2. DOUBLE-DIFFERENTIAL

NEUTRON

MULTIPLICITIES

FOR 96Zr+96Zr

Apart from shell effects in the precompound neutron multiplicity, the energy and angular distributions of the emitted neutrons are of specific interest. Simple inspection of the continuum density p,(r) (3.6) (cf. fig. 4) yields a first estimate upon the angular distribution. It is seen to be essentially forward-backward peaked, but no information on the velocity or energy distribution may be gained in this way. In order to evaluate the double-differential neutron multiplicity d2MM,/dEd8, either a Fourier-Bessei transfo~ation of the continuum wave functions may be applied or

W. Gassing / Diabatic neutron emission

269

2.0

1.0

{m

[ MeV”‘1

Fig. 6. Precompound neutron multiplicities for 90Zr+90Zr and 96Zr+96Zr as in fig. 5, plotted as a function of the relative velocity above the corresponding Coulomb barriers. The straight lines are fitted to the numerical “data” in order to obtain the thresholds for diabatic neutron emission.

a Fourier analysis with respect to their energy components for large IrI ref. 26). The latter technique has proven to be more successful and is performed in close analogy to ref. 26), where a related one-dimensional problem has been studied. For the evaluation of the double-differential neutron multiplicity the Schriidinger equation (2.28) is integrated on a grid [0,18] fm x C-35351 fm with 51 x 161 meshpoints in p and z in constant time steps of 10m2* s. The time integration is stopped at t = 1.5 x 10m2’ s, which introduces a slight uncertainty in the neutron spectra for energies E < 15 MeV but leaves the shape at higher energies unchanged. Simultaneously with the time integration the wave functions are Fourier-transformed with respect to time at 25 fm < It-1< 28 fm. In order to avoid contributions to the spectrum from reflection at the boundary of the finite grid, the wave functions are suppressed by a smooth cutoff factor for lzl > 30 fm and p > 16 fm. It is found that the quantity M,(E, 8; r) = r2(2E/M)f(2nh)-2

s

1 n,(t,) Y

x I dt$,(r, 0; tkw {(W)Et)12

(3.9)

is almost independent of I for r > 25 fm. In eq. (3.9) cylindrical coordinates p,z have been rewritten in terms of spherical coordinates I, 8 in obvious notation. It is easily verified that M,(E, 0) is identical with the double-differential neutron multiplicity d2M,/dEd0 except for a constant factor 26). The distribution function

270

M,(E,

W.

0.

r)

is finally averaged

Cussing

,! Diahuric

neutron

emission

over r for fixed 0. i.e.

s r,+Ar

M,(E, 0) = (lldr)

1.

(3.10)

drM,(E, 0, r)

with ri = 25 fm and Ar = 3 fm. The rather (CPU) time-consuming procedure yields energy and angular distributions with numerical errors less than 50% up to energies of 60 MeV. Any further improvement of the accuracy was found to be accompanied by a tremenduous increase in computer time without, however, changing the shape of the spectra. The actual results for M,(E, 0) (3.10) are displayed in fig. 7 for central collisions of Y6Zr+96Zr at 12 MeV/u and 20 MeV/u. The straight lines, drawn to guide the eye, indicate that the tails of the spectra are compatible with an exponential slope, which is roughly constant as a function of the angle 0. Surprisingly, this slope is approximately the same for 12 MeV/u as for 20 MeV/u. When fitting a Boltzmann distribution d’M,/dEdO

y EfT;

i

exp ( - E/T,}

; I. NE:TRON

I. 20

I * I. ENERGY 30 40

1, I 50(MEW 60

(3.11)

20 MEV/U 10

20

NEUTRON

30

40

ENERGY

50

60

(MEW

Fig. 7. Double-differential neutron multiplicities (3.10) for central collisions of 90Zr+9bZr at 12 MeV/u and 20 MeV/u in arbitrary units, The straight lines are drawn to guide the eye.

W. Gassing 1 Diabatic

neutron emission

271

to the high-energy tails of the distribution, a temperature parameter To = 7.6kO.8 MeV may be extracted. The latter value is compatible with the slope obtained in the semiclassical calculation 35) for 92Mo -t9’Mo at 12 MeV/u, while at 20 MeV/u a difference of almost 2 MeV in the slope parameter is found. This actually limits the applicability of the semiclassical approach 35) to bombarding energies less than 15 MeV/u [cf. ref.48)]. The angular distribution as obtained from integrating M,(E,O) (3.10) over energy is shown in fig. 8 for 96Zr+96Zr at 12 MeVJu and 20 MeV/u. Due to the symmetry of the reaction M,(0) is identical to M,(180°-8). The enhancement of the neutron yield in forward (backward) direction is quite significant and increases with bombarding energy. It should be noted that the spectral distribution of neutrons emitted in coincidence with central collisions of 90Zr + 90Zr is the same as for 96Zr + 96Zr at the corresponding lab energies. The only difference found within the accuracy of the calculation is the total increase of the precompound neutron multiplicity with the neutron excess (cf. eq. (3.1). Up to now, neutron emission from heavy symmetric systems has not been studied experimentally in the interesting energy range above 10 MeV/u. The experiments performed with light projectiles like i2C, 13C, I60 and “Ne on heavy targets like Sm, Gd, Ho or U are too asymmetric to be investigated within the two-center shell model (2.7). Any comparison with experiment thus has to be taken with great care. Nevertheless, some numbers from the “Ne + 165H~ reaction at 11, 14.6 and 20 MeV/u [ref. I”)] are quoted in order to show the general trend of the nonequilibrium neutron data. This system leads to a similar compound system as the symmetric ones discussed here. Furthermore, the neutrons have been detected in coincidence with evaporation residues 16). This is supposed to represent a I

dM, /d@cM 1.5 -

-

20 MeVlu

- ---

12 MeVlu

30-

lo-

I

so0

&M Fig. 8. Angular distributions in the center-of-mass system resulting from diabatic neutron coincidence with central collisions of “%Zr + 96Zr at 12 MeV/u and 20 MeV/u.

emission

in

272

W. Gassing / Diabatic

neutron emission

convenient trigger for almost central collisions of heavy systems. The multiplicity of the precompound neutron component was found to be 0.4 for 11 MeV/u, 1.4 for 14.6 MeV/u and 1.5 for 20.1 MeV/u, while the slope parameters T, were determined by a moving source fit to be 4.5 MeV, 6.3 MeV and 8.6 MeV, respectively. Such an increase of the slope parameter is not found for the diabatic emission process, though the latter values are compatible with T, = 7.61LO.8 MeV from 15 MeV/u to 20 MeV/u. Since any extraction of slope parameters depends on the velocity of the moving source and a proper substraction of the equilibrium component, we may only conclude that the neutron spectra are not in complete disagreement with data for asymmetric systems. The increase of the multiplicity with energy is additionally of the right order of magnitude, if we take into account the different sizes of the projectiles and the corresponding excitation energies of the compound-like system (cf. fig. 5). Summarizing this sect., we find a preequilibrium neutron component in central collisions of heavy symmetry systems, which is roughly exponential in the highenergy tails with slope parameters T, in the order of 7.6kO.8 MeV for lab bombarding energies between 12 MeV/u to 20 MeV/u. The angular distribution of the emitted neutrons is considerably forward-backward enhanced in contrast to thermal emission processes. Since diabatic neutron emission is related to nonequilibrated coherent nuclear motion, shell effects still show up in the total multiplicity for neighbouring isotopes, a significant effect especially when stepping from 90Zr+ 90Zr to g6Zr+96Zr.

4. Comparison with other approaches The emission of neutrons in central heavy-ion reactions due to the diabatic shift of s.p. levels is based on the assumption that the mean free path Iz of a nucleon is not significantly smaller than the diameter of the nuclear system. This limits the applicability of the diabatic approach to lab energies less than 17 MeV/u for symmetric systems like 96Zr + 96Zr [ref. ““)I, if I is evaluated from eq. (2.36). Due to an estimated uncertainty of 30% in 1 the upper limit for diabatic s.p. motion may extend to 20 MeV/u bombarding energy. This implies that the formation of a local hot zone is rather unlikely in this energy domain. The diabatic approach and the hot-spot models 31*32) thus are conceptually in opposition to each other. The relation to other nonequilibrium models is studied more closely in the following subsections. 4.1. THE

FERMI-JET

MODEL

The Fermi-jet approach 25-28) is based on the idea that nucleons are exchanged between the colliding nuclei as soon as the time-dependent barrier between the

W. Cussing / Diabatic

nuclei vanishes (cf. fig. 1). In the limit of independent velocity u, in the donor nucleus A, has a velocity Ub

=

273

neutron emission

s.p. motion a nucleon with

u,+d

(4.1)

in the moving frame of the recipient nucleus A, where & denotes the relative velocity at the time of contact. The kinetic energy of the nucleon in the local frame of A, then is given by Eb =

@41u,12.

(4.2)

The gain in momentum of the nucleon in the recipient nucleus is compensated by a corresponding loss of relative momentum. The fate of the further nucleon trajectory is determined at the surface of the recipient, more precisely, by the height of the time-dependent barrier U,, and its velocity u,. The kinetic energy of the nucleon with respect to the moving barrier is determined by E,

=

$l4Iu,+lk-u,~*

(4.3)

and the nucleon will be emitted into the continuum, if E, 2 U,. The neutron spectra resulting from corresponding trajectory calculations are highly forwardbackward peaked 25*28*60), since the boost in transvers directions is very small. The effect of the two-body collisions is taken into account by an attenuation factor (4.4) for the nucleon flux from the donor to the recipient nucleus and vice versa. In eq. (4.4) the length d is the distance traversed by the fast particle in the recipient nucleus and 1 denotes the mean free path. Due to energy and momentum conservation scattered nucleons are energetic, too, and may overcome the barrier as well (two-body PEPS [ref. *‘)I. Their contribution to the continuum, however, was found to be less than 30% for 12C+ “‘Cd up to bombarding energies of 20 MeV/u [ref. ““)I. In order to compare the Fermi-jet concept with the diabatic approach, the Schriidinger equation (2.28) is integrated for states tiy which are initially located in nucleus A, for z > 0. The continuum density for (#+z*)+ otherwise,

then yields information

about the corresponding

c 12 fm

quantum-mechanical

(4.5)

emission

214

W. Cussing 1 Diabaiic

neutron

emission

96 ZR + 96 ZR (17.5 MEVAJ)

t=70

ES

Fig. 9. Contour diagram of the continuum density p: (4.5) at r = 7 x lo-*’ s for a central collision of 96Zr+g6Zr at 17.5 MeV. The density increases by a factor of 2 from line to line.

pattern for t + co. The latter quantity is shown in fig. 9 at t = 7 x 10-22s for a central collision of 96Zr+96Zr at 17.5 MeV/u. The density contours increase by a factor of 2 from line to line. Two components may be distinguished in fig. 9: (i) a rather fast component emerging in the -z direction, and (ii) a slow component emitted with a slight time delay in opposite direction. Integrating & over p and z for t --+00, the component (i) is found to contribute by more than 90% to the density in the continuum. It clearly represents a quantum-mechanical analogue of Fermi jets. Thus the angular distribution resulting from diabatic neutron emission is very close to the angular distribution from Fermi-jet calculations, both showing a drastic enhancement in the forward (backward) direction 25S28*60) (cf. fig. 8). The qualitative agreement between the two approaches, however, is not surprising because the Wigner transform6r) of the one-body density matrix P,Jr, k; t) = (1/(2~)~) c

exp{ik. s}t,by*(r+$s, t)

Y x

l&(r-4s; t)d3s

(4.6)

obeys the Vlasov equation 62) in the classical limit 63, 64). The solution of this equation e.g. may be constructed by tracing the corresponding classical trajectories of all initial phase-space elements in time65). The treatment of two-body collisions is slightly different in the diabatic and Fermi-jet approach because the effective mean free path A, for a state Ii/,(t) is explicitly time-dependent via cy(t) (cf. eq. (2.36)). The continuum emission from scattered nucleons (two-body PEP’s [ref. ““)I), furthermore, is neglected in the present diabatic treatment and is considered to be a major shortcoming. On the other hand, the Fermi-jet model assumes nuclear momentum distributions as given by the Fermi-gas limit. Numerical studies of the respective one-body phase-space distributions, however, show significant distortions 64) for overlapping nuclei,

W. Cussing 1 Diabatic neutron emission

27s

which in turn modify the neutron spectra in the high-energy tail considerably. Such distortions are obviously included in the present diabatic approach since the neutron spectra are evaluated from appropriate time-dependent two-center shellmodel states $,(t). We finally find that diabatic neutron emission may be considered as a quantummechanical analogue to the Fermi-jet models25-2*). In contrast to TDHF, where corresponding fast-particle jets are known for some time 33), it allows for the evaluation of double differential multiplicities d’M,/dEdB due to the linearity of the Schriidinger equation (2.1). 4.2. THE EXCITON

MODEL

The Griffin-exciton model 19) and the Harp-Miller-Berne model 20) were originally proposed for p- and a-induced reactions and have been applied to heavyion reactions in recent years, too 2’-23). The basic assumption of, the models is that a set of eigenstates of an approximate hamiltonian for the target nucleus can be subdivided into classes of proton and neutron particles and holes with respect to the ground state r9). The incident ion is viewed to produce an excited state which is characterized by an initial distribution of particles and holes. This initial distribution, furthermore, is assumed to relax by residual two-body collisions leading ultimately to the equilibrium distribution of the average compound nucleus. Since the matrix elements of the residual interactions are taken to be constant, the equilibration is dominated by the phase-space for the transitions, i.e. by the available number of levels for given initial number of excitons at fixed total excitation energy. As the nuclear mean field of a heavy target is roughly constant in time for pand m-induced reactions, the total phase-space may be divided into energy bins of 1 MeV. The initial particle-hole distribution (exciton number no) with respect to the Fermi level, will change in time either due to the coupling to the continuum or due to residual two-body collisions. The time dependence of the occupation numbers is determined by a master equation with transition rates proportional to free nucleon-nucleon cross sections 21). The model then involves two parameters, (i) the strength of the transition rates, and (ii) the initial exciton number no. While the first parameter reflects the average mean free path of nucleons within nuclei and is adjusted to a value of four times the mean free path of nucleons in nuclear matter 16b), the initial exciton number no is found to be the decisive parameter with respect to energy spectra 19-23* 60). For light-ion-induced reactions no is given by the total number of nucleons in the projectile independent of the bombarding energy. The latter relation, however, no longer holds for 12C or ‘ONe induced reactions r6, 60) and no has been treated as a free parameter. Obviously, the question arises about the physical meaning of no in case of heavy-ion collisions. The diabatic two-center shell model allows for a partial answer to this question,

216

W. Cussing 1 Diabatic

neutron emission

since the time dependence of s.p. levels E.,(C)is evaluated explicitly via eq. (2.42). It is found numerically that the matrix elements of the finite two-center hamiltonian H (2.26) between the time-dependent states IL,,(t) from eq. (2.27) and (2.28) are roughly identical to the matrix elements of the infinite two-center hamiltonian H, (2.7) between the corresponding travelling diabatic s.p. states #?(q,tj) (2.16) when correcting fo’r the finite depth of the potential V, (2.25) i.e.

~~(0= (IL,(OIH(q(O)ll(/,(O) 2

(&h 4W,(qWM,(q, 4 )>-

v,

= 9%MfMn+&,(q)- v,

(4.7)

Eli(q) = w~,(qw%l)~ w%Jl4,(qb

(4.8)

with

The relation (4.7) holds in the initial phase of the collision4*) where a large overlap of the nuclei is achieved on a short time scale and the coupling to the continuum is still of minor importance. In the limit of a single collective degree of freedom R, as considered in sect. 3, eq. (4.7) approximately reads4’*48)

v, Ey(Oz $+fd2(R)+~y(~)-

(4.9)

for central collisions of symmetric systems, where d(R(t)) is determined by the effective equation of motion (2.39). It should be noted that the collective energy per nucleon M/8d2 is less than 4 MeV in the energy range investigated. The distribution of the diabatic s.p. levels cy(R,) at the classical turning point defined by d(R,) = 0 then directly yields the number of excited states above the local Fermi level +(R). This number may easily be extracted from diabatic level schemes as displayed in fig. 3. Please note again that each level in fig. 3 corresponds to two states because of time reflection symmetry and that proton levels above the corresponding Fermi level have to be counted as excitons as well. A more systematic study will be presented in a forthcoming publication 66). The direct coupling of the s.p. degrees of freedom to the collective motion, as described by diabatic s.p. motion (cf. sect. 2), yields a simple and intuitive picture (fig. 3) with respect to the creation of excited states (excitons). The decay of the excited states due to the coupling to the continuum or due to residual two-body collisions may be described by a master equation as in refs. 1g-23*s*) or by a relaxation ansatz as in ref. “). Consequently, the diabatic approach and the exciton model are conceptually similar if (i) one determines the number of initial excitons from the diabatic level scheme at the turning point, and (ii) one reduces the sixdimensional one-body phase-space to a single dimension, i.e. the s.p. energy above the Fermi level. The underlying s.p. hamiltonian, however, is explicitly timedependent in contrast to the exciton model.

W. Gassing / Diabatic

neuiron emission

277

In the present treatment of diabatic neutron emssion, the further creation of “secondary excitons” by two-body collisions has been disregarded. This neglect definitely needs improvement. On the other hand, the coupling of the “primary excitons” to the continuum is treated quantum-mechanically. The resulting double differential neutron multiplicities are found to be very close to neutron spectra evaluated from Fermi-jet models 25, 28), indicating th at the diabatic approach yields a transparent link between the concepts.

5. Discussion The precompound emission of neutrons in central nucleus-nucleus collisions has been studied within the framework of dissipative diabatic dynamics for a timedependent finite two-center shell-model hamiltonian. The energy spectra show an exponential tail with a slope parameter roughly independent of the angle 0. The angular distributions are significantly enhanced in the forward (backward) direction. Furthermore, the total precompound neutron multiplicity is found to be quite sensitive on the neutron excess of the Zr isotopes considered. This effect may be traced back to the respective shell structure of the nuclei. The extensive calculations performed can be understood in a rather simple picture. Due to the direct coupling of the s.p. motion to the collective degrees of freedom highly excited states are produced which store the collective kinetic energy during the initial phase of the collision. Such excited states may be considered as initial excitons within the familiar exciton models 1g-23), if one reduces the sixdimensional one-body phase-space to a one-dimensional quantity, i.e. the particlehole distribution with respect to the Fermi level. The excitons couple to the continuum or to more complicated states. Yet the quantum-mechanical coupling of the highly excited states to the continuum yields neutron spectra, which are roughly exponential in energy as in the exciton models, but considerably forward (backward) enhanced as in the Fermi-jet approaches 2s-28). This is not surprising because the quantum-mechanical emission process may alternatively be evaluated within the one-body phase-space, where the Vlasov equation62) is known to describe the time evolution of the corresponding classical phase-space density a3-65). In this respect, the diabatic approach can be considered as a quantum-mechanical analogue of the Fermi-jet model. In contrast to present selfconsistent TDHF calculations 33), shell effects in the s.p. level density can be included in the diabatic approach without major difftculties. Furthermore, the linearity of the s.p. Schriidinger equation allows one to analyse the emitted neutron density with respect to energy and scattering angle. A major shortcoming of the present diabatic approach is the approximate treatment of residual two-body collisions (subsect. 2.3). While the decay of the excited states is taken into account, the emission from continuum states, which are

27X

W’. Cussing : Diabarrc mwron

rmissron

populated by two-body collisions, is neglected. This definitely needs improvement. The Vlasov equation with a collision term of Uehling-Uhlenbeck type”) can be regarded as a natural generalization body phase-space. First solutions

of the present diabatic approach in the onewith respect to light-particle emission are in

progress 64. 6*). In spite of the clear statements with respect to characteristic properties of precompound neutron emission in the diabatic approach. a few comments appear to be necessary. First of all. the preequilibrium neutron emission observed in central heavy-ion collisions contributes only 5 3, to 15 “,, to the total neutron multiplicity in the energy range from IO MeV/u to 20 MeVju [ref.6n)]. Furthermore. the calculated multiplicities do not account for the to!al preequilibrium neutron multiplicity because contributions from nucleon-nucleon scattering are disregarded. Since shell effects in the total precompound neutron multiplicity are maximal for symmetric systems. rather high excitations energies of the compound-like system are involved. The temperature of the equilibrated nuclear system is found to be close to the apparent temperature parameter T, extracted from the precompound neutron or proton spectrah9). This induces a rather large uncertainty in the separation of preequilibrium and equilibrium light-particle multiplicities. Despite the experimental difficulties. the shell effects in the precompound neutron multiplicity as discussed in subsect. 3.1 remain interesting. They should be observed in central collisions of 90Zr +90Zr, 96Zr+96Zr or 92Mo+92M~ and ‘“MO + 98Mo in the high-energy tails of neutron spectra in the forward (backward) direction when increasing the bombarding energy from 10 MeV/u to 20 MeV/u. The author acknowledges valuable discussions D. Hilscher. J. Knoll. H. Machner. W. Norenberg.

with U. Brosa. C. K. Gelbke. F. Plasil. W. U. Schroder. and

D. Sperber.

References I) H. C. Britt and A. R. Quinton. 2) R. Btmbot,

D. Gardes

Phys. Rev. 124 (1961)

and M. F. Rivet. Nucl.

877

Phys. Al89

3) J. W. Harris, T. M. Cormier. D. F. Geesaman. Phys. Rev. Lett. 38 (1977) 1460

(1972)

193

L. L. Lee, Jr., R. L. McGrath

4) C. K. Gelbke. M. Bini, C. Olmer, D. L. Hendrte, J. L. Lavtlle. D. K. Scott and H. H. Wieman. Phys. Lett. 716 (1977) 83 5) H. Ho. R. Albrecht. and F. Scheibling. 6) Y. Eyal, A. Gavron.

W. Diinnweber.

Cl. Graw.

Z. Phys. AZ83 (1977) I. Tserruya.

S. G. Steadman.

and J. P. Wurm,

J. Mahoney.

J. P. Wurm.

M.

C.

D. Disdier,

Mermaz. V. Rauch

235

Z. Fraenkel.

Y. Eisen. S. Wald,

R. Bass. C. R. Gould,

G. Keyling,

R. Renfordt. K. Stelzer, R. Zitzmann. A. Gobbi. U. Lynen, H. Stclzer. I. Rode and R. Bock. Phys. Rev. C21 (1980) 1377; Phys. Rev. Lett. 41 (1978) 625 7) D. Hilscher, J. R. Birkelund. A. D. Hoover. W. U. SchrMer. W. W. Wilcke. J. R. Huizenga. A. C. Mignerey, K. L. Wolf, H. F. Breuer and V. E. Viola. Jr., Phys. Rev. C20 (1979) 576 8) B. Tamain, R. Chechik. H. Fuchs, F. Hanappe, M. Morjean. C. Ngo. J. Peter. B. Lucas. C. Mazur, M. Ribrag and C. Signarbieux, Nucl. Phys. A330 (1979) 253

M.

Dakowski.

W. Gassing 1 Diabatic neutron. emission

279

9) C. R. Gould, R. Bass, J. v. Czamecki, V. Hartmann, K. Stelzer, R. Zitzmann and Y. Eyal, Z. Phys. A294 (1980) 323 10) M. Bini, C. K. Gelbke, D. K. Scott, T. J. M. Symons, P. Doll, D. L. Hendrie, J. L. Laville, J. Mahoney, M. C. Mennaz, C. Olmer, K. Van gibber and H. H. Wieman, Phys. Rev. C22(1980) 1945 11) A. Gavron, J. R. Beene, R. L. Ferguson, F. E. Obenshain, F. Plasil, G. R. Young, G. A. Petitt, K. G. Young, M. Yalskellinen, D. G. Sarantites and C. F. Maguire, Phys. Rev. C24 (1981) 2048 12) J. Kasagi, S. Saini, T. C. Awes, A. Galonski, C. K. Gelbke, G. Poggi, D. K. Scott, K. L. Wolf and R. L. Legrain, Phys. Lett. 104B (1981) 434 13) R. P. Schmitt, G. J. Wozniak, G. U. Rattazzi, G. J. Mathews, R. Regimbart and L. G. Moretto, Phys. Rev. Lett. 46 (198 1) 522 14) M. 9. Tsang, W. G. Lynch, R. J. Puigh, R. Vandenbosch and A. G. Seamster, Phys. Rev. C23 (1981) 1560 15) T. C. Awes, S. Saini, G. Poggi, C. K. Gelbke, D. Cha, R. Legrain and G. D. Westfall, Phys. Rev. C25 (1982) 2361 16) E. Holub, D. Hilscher, G. Ingold, U. Jahnke, H. Orf and H. Rossner, Phys. Rev. C28 (1983) 252; E. Holub, M. Korolija and N. Cindro, Z. Phys. A314 (1983) 347 17) C. K. Gelbke, Nucl. Phys. A408 (1983) 473~ 18) A. Gavron, J. R. Beene, 9. Cheynis, R. L. Ferguson, F. E. Obenshain, F. Plasil, G. R. Young, G. A. Petitt and C. F. Maguire, Phys. Rev. C26 (1983) 450 19) J. J. Griffin, Phys. Rev. Lett. 17 (1966) 478; Phys. Lett. 249 (1967) 5 20) G. D. Harp, J. M. Miller and 9. J. Berne, Phys. Rev. 165 (1968) 1166; G. D. Harp and J. M. Miller, Phys. Rev. C3 (1971) 1847 21) M. Blann, Ann. Rev.Nucl. Sci. 25 (1975) 123; Phys. Rev. C23 (1981) 205 22) H. Machner, Phys. Rev. C21 (1980) 2695; Z. Phys. A302 (1981) 125; Phys. Rev. C28 (1983) 2173 23) K. Niita, Z. Phys. A316 (1984) 309 24) D. H. E. Gross and J. Wilczynski, Phys. Lett. 67B (1977) 1 25) J. P. Bondorf, J. N. De, G. Fai, A. 0. T. Karvinen, 9. Jakobson and J. Randrup, Nucl. Phys. A333 (1980) 285 26) U. Brosa and W. Krone, Phys. Lett. 105B (1981) 22 27) H. Tricoire, Z. Phys. A312 (1983) 221; A317 (1984) 347 28) F. Sebille and 9. Remaud, Nucl. Phys. A420 (1984) 141; K. T. R. Davies, B. Remaud, M. Strayer, K. R. Sandya Devi and Y. Raffray, Ann. of Phys. 156 (1984) 68 29) L. Zybert, A. Budzanowski and R. Zybert, Phys. Rev. C25 (1982) 3201 30) F. Redish and D. M. Schneider, Phys. Lett. 1OOB (1981) 101 31) R. Weiner and M. Westrom, Nucl. Phys. A286 (1977) 282; P. A. Gottschalk and M. Westrom, Nucl. Phys. A314 (1979) 232 32) W. W. Morison, S. K. Samaddar, D. Sperber and M. Zielinska-Pfabe, Phys. Lett. 93B (1980) 379; P. Mooney, W. W. Morison, S. K. Samaddar, D. Sperber and M. Zielinska-Pfabe, Phys. Lett. 98B (1981) 240 33) H. Stocker, R. Y. Cusson, J. A. Maruhn and W. Greiner, Phys. Lett. 1OlB (1981) 379; K. R. S. Devi, M. R. Strayer, K. T. R. Davies, S. E. Koonin and A. K. Dhar, Phys. Rev. C&l (1981) 2521; A. K. Dhar, M. Prakash, K. T. R. Davies, J. P. Bondorf, 9. S. Nilsson and S. Shlomo, Phys. Rev. C25 (1982) 1432 34) W. Cassing, A. Lukasiak and W. Norenberg, Proc. Int. Workshop on gross properties of nuclei and nuclear excitations XI, Hirschegg, 1983, p. 107 35) W. Cassing and W. Norenberg, Nucl. Phys. A431 (1984) 558 36) J. W. Negele, Rev. Mod. Phys. 54 (1982) 913 37) H. Flocard, Nucl. Phys. A387 (1982) 283~ 38) P. Morel and P. Nozieres, Phys. Rev. 126 (1962) 1909 39) M. T. Collins and J. J. Griffin, Nucl. Phys. A348 (1980) 63 40) G. F. Bertsch, Z. Phys. A289 (1978) 103 41) W. Cassing, A. K. Dahr, A. Lukasiak and W. Norenb-erg, Z. Phys. A314 (1983) 309 42) M. Prakash, S. Shlomo, 9. S. Nilsson, J. P. Bondorf and F. E. Serr, Nucl. Phys. A385 (1982) 483

280

W. Cussing / Diabatic neutron emission

43) K. K. Kan and J. J. GrifIin, Phys. Rev. Cl5 (1977) 1126; Nucl. Phys. A301 (1978) 258 44) H. C. Pauli and L. Whets, Z. Phys. A277 (1976) 83; D. Glas, Z. Phys. A277 (1976) 195 45) A. Lukasiak, W. Cassing and W. Niirenberg, Nucl. Phys. A426 (1984) 181 46) P. Holzer, U. Mosel and W. Greiner, Nucl. Phys. Al38 (1969) 241; D. Scharnweber, W. Greiner and U. Mosel, Nucl. Phys. Al64 (1971) 257; J. Maruhn and W. Greiner, Z. Phys. 251 (1972) 431 47) P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980) p. 76 48) W. Cassing and W. N&e&erg, Nucl. Phys. A433 (1985) 467 49) W. Cassing and W. N&et&erg, Nucl. Phys. A401 (1983) 467 50) W. D. Myers, Droplet model of atomic nuclei (IFI, Plenum, NY, 1977) 51) A. H. Wapstra and K. Bos, At. Nucl. Data Tables 19 (1977) 215 52) D. Berdichevsky, unpublished 53) W. Norenberg, Phys. Lett. 104B (1981) 107 54) U. Brosa and W. Cassing, Z. Phys. A307 (1982) 167 55) J. J. Griffm, Invited lectures presented at the RCNP Summer School for Nuclear Physics, Kyoto, Japan, May 1983 56) B. Strack and J. Knoll, Z. Phys. A315 (1984) 249 57) K. Cc-eke and P. G. Reinhard, ed., Time-dependent Hartree-Fock and beyond, Lecture Notes in Physics, vol. 171 (Springer, Berlin, 1982) 58) S. Ayik and W. N&e&erg, Z. Phys. A309 (1982) 121; S. J. Wang, Phys. Lett. 133B (1983) 27; P. Buck and H. Feldmeier, Phys. Lett. 129B (1983) 172; S. Ayik, Nucl. Phys. A370 (1981) 317; A422 (1984) 331; H. Reinhardt, R. Bahan and Y. Alhassid, Nucl. Phys. A422 (1984) 349 59) R. Bass, Proc. Symp. on deep-inelastic and fusion reactions with heavy ions, Lecture Notes in Physics 117, ed. W. v. Oertzen (Springer, Berlin, 1980) p. 281 60) D. Hilscheri E. Holub, G. Ingold, II! Jahnke, H. Orf, H. Rossner, W. U. Schroder, H. Gemmeke, K. Keller, L. Lassen and W. Lucking, Invited talk presented by D. Hilscher at the Workshop on coincident particle emission from continuum states, Bad Honnef, June 67, 1984, preprint HMIP-84/3R 61) E. Wigner, Phys. Rev. 40 (1932) 749 62) A. Vlasov, J. Phys. (USSR) 9 (1945) 25 63) J. E. Moyal, Proc. Camb. Phil. Sot. 45 (1949) 99 64) W. Cassing, Proc. XXII Int. Winter Meeting on nuclear physics, Bormio, Italy, Jan. 1984, p. 342 65) C. Y. Wong, Phys. Rev. C25 (1982) 1460 66) W. Cassing, to be published 67) E. A. Uehling and G. E. Uhlenbeck, Phys. Rev. 43 (1933) 552; H. Mori and S. Ono, Prog. Theor. Phys. 8 (1952) 327 68) C. Gregoire, F. Scheuter, B. Remaud and F. Sebille, Proc. XXII Int. Winter Meeting on nuclear physics, Bormio, Italy, Jan. 1984, p. 209 69) N. Herrmann, S. Gralla and A. Gobbi, private communication