Nuclear F'hysics A428(1984)145c-16oC North-Holland,Amsterdam

145c

OPEN QUESTIONS: THERMALIZATION AND FLOW, KINETIC OR POTENTIAL DRIVEN?

James J. GRIFFIN and Wojciech RRONIOWSKI

Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742 U.S.A.

The fact that neutron drifts seem not to follow the expectations based on energetically dominated statistical descriptions of the deep inelastic reactions is welcomed as keeping open the hope that specific mechanistic aspects of the nuclear flow and thermalization processes might yet emerge from beneath the cover of these reactions' statistical complexity. A one- dimensional SchrGdinger DOUBLE WELL model is analyzed to show that its Schrodinger dynamical flow does not conform to the common statistical assumption of total energy dominance. Rather, nucleonic kinetic energy emerges as a stronger determinant of the early flow than potential energy. Since these "kinetic pressures" are very sensitive to the radii of the nucleonic potential wells, their study promises to illuminate the question of differences between the neutron and proton radii. Also, for kinetic pressure dominated exchanges, one-dimensional loci of equilibrium points in the projectile-like (Z,N) plane replace the isolated stationary points of the energy surface. One thus arrives at a two-stage equilibration process,in which the early rapid kinetic pressure driven drift toward a kinetic pressure equilibrium is followed by a slower potential,driven drift along the locus of kinetic equilibria towards the energy equilibrium point. Were the neutron and proton loci of equilibria to separate, an "equilibrium channel" could arise, which would tend to guide the N-Z exchanges in a direction which might even be opposite to the preference of the overall energy.

1. TWO MAJOR PHYSICAL QUESTIONS STILL OPEN FOR NUCLEAR HEAVY-ION PHYSICS

Two large questions which nuclear heavy ion experimentation is uniquely fitted

to illuminate concern the fluidics of Fermi-degenerate nuclear matter, and the

process of evolution from a zero-temperature pure quantum state into a state

sufficiently complicated to warrant description in terms of a statistical mixed

state density matrix. Since both the flow and the thermalization of the

initially simple nuclear state are realized during the deep inelastic heavy ion

reactions, the data from such processes mrst provide cur prime inferential basis

on these questions.

?Work supported by U. s. Department of Energy and University of Maryland Computer Science Center.

03’?‘5-9474/84/503.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Among such data, the distributions in the N and Z values, the emergent angles,

and final kinetic energies of fragments from these reactions have become in

recent years the object of increasingly detailed experimental and theoretical

study. From initial measurements restricted to the Z-distributions vs. the

emergent fragment energies, experiments have evolved to provide two dimensional

,(N,Zf distributions vs. the total kinetic energy loss (TKEL), as well as other

information about the fragments, such as angular momentum information based on

y-ray studies.

2. STATISTICAL ~ES~RIPTIQ~S OF (NJ) #ISTRI8~TI~~S

Apace with these experimental developments. theoretical descriptions were

fashioned to encompass this new data, beginning with the Fokker-Planck descrip-

tion of Niirenberg et al.I, and the phenomenologically elaborate moment analyses

of Randrup et al. 2 In particular, the latter provided remarkable agreement with

the measured values of various widths of the fN,Zf distributions. However, a

somewhat analogous3p4 discrete Random Halk treatment based on a few very simple

physical assumptions, while it showed that these widths were essentially a

two-parameter phenomenon, at the same time was unable to reproduce their behavior

at small TKEL values from any of three simple statistical assumptions utilizing

specified final state level densities to fix the transition probabilities.5.

Both the moment analysis and the discrete Randy Halk descriptions incorporate

in an essential way the dinuclear ground state energy surface as an essential

determinant of the evolution of the (N,Z) distribution. In the moment analysis,

derivatives of this dinuclear energy provide the "driving forces" which influence

the drift and diffusion coefficients of the equation. In the Random Walk

approach level densities (of whichever type) are calculated with respect to the

final configuration~s ground state energy, as defined by such a dinuclear energy

surface. Thus into both descriptions are structured strong tendencies for the

distribution to evolve most rapidly along lines of steepest dinuclear energy

descent towards the valley and the minima of the dinuclear energy.

J. J. Griffin, W. Broniowski / Open questions : the~itzation and flow 147c

As an example, the dinuclear energy surface for s6Fe+l%o is presented in

Fig. 1. Any description under which the (N,Z) distributionevolves according to

the ground state energetics which this surface describes must predict that the

ssFe projectile should tend simultaneously to lose protons andtogain neutrons

from its 165~0 partner leading to average values of (N,Z) following the arr[M

marked ~STATISTICAL FZlDEL FRE~ICTION~ in Fig. 1. Indeed, the observedS*Z proton

drift plotted in Fig. 2 as Z vs. TKEL agrees qualitatively with this predic-

tionft. However, the neutron flow contravenes this energetic tendency, even

r....r...,r....r....I...l 20 30 40

NP

FIGURE 1

DAYA; CORRECTED FOR N-EMISSION by BRE”ER,

30

EL (MeV)

2.6

RANDOM WALK

f 24

DATA 22

EL (MeV)

FIGURE 3 FIGURE 4

148~ J.J. Griffin, W. Broniowski / Open questions : thermalization and fiow

after the datahave been (significantly) corrected7 for prompt neutron emission,

as shown in Fig. 3. Very similar qualitative discrepancies between the n-emis-

sion corrected neutron drift and the implications of the corresponding dinuclear

potential energy surfaces occur also in the reactions6.7, seFe+*osBi and

ssFe+*sau. In the reaction, szCr+20gBi, the discrepancy8 is even more striking

in that after corrections for neutron emission the value of i for the Q-like

fragment is not found to remain roughly constant as inthereactionsmentioned

above, but persists in decreasing (by _ 3.5 units at TKEL q 100 MeV) in flat

contradiction with the implication of the dinuclear energy surface, on which

downhill motion implies an increase in N.

We believe that these discrepancies, qualitative as they are, warrant a

thorough review of the assumptions underlying the statistical descriptions of

these processes. Moreover, this development is likely in our view to be a

positive one, as we ncwr briefly mentiong.

3. DISCREPANCIES WITH STATISTICAL THEORIES MAY ALLOW DEEPER PHYSICAL PROGRESS

To return to the central fundamental questions of nuclear "Fermidynamical

flow" and the "thermalization" of the initially pure heavy ion state, we empha-

size that the phenomenological development of the theoretical side has ledusto

descriptions which suppress dynamical quanta1 details in favor of statistical

tendencies determined primarily by the overall energetics of the process. Were

the phenomenological success of Randrup et al.2 to extendoverthewholerange of

heavy ion data, then access to these deeper questions might ultimately be frus-

trated by the difficulty of identifying for them definitive signaturesamongthe

observed data. Indeed, even the present partial success of that work has led

some experimentalists to abandon further study of (N,Z) distributions on the

grounds that they are already well comprehended by the theory, and therefore

unlikely to lead to novel insights, in spite of the fact that the "theory" here

depends upon so many umproven assumptions and incorporates a range ofphenomeno-

logical nuclear properties whose separate proportionate influence on the predic-

tions remains untraced. On the other hand, one nust recognize also the more

pessimistic possibility that these deeper questions may be obscured beyond the

reach of both our current experimental and theoretical probes by the very

complexity of the process itself.

From this viewpoint, it is a good turn of fortune if our pheno~nological

success is circumscribed by qualitative failures. Then the hope may yet persist

that further careful study will finally reveal truths from theseprocesses deeper

than the statements that they are simply statisticalandenergetically dominated.

Thus, one can hope that in order to understand fully the observeddatawewill be

forced to analyze,to at least sane modest level of specificity, the interplay

between the exact time reversible Schradinger equation's description which we

presume to be fundamentally true, but practically too complicated, and the

time-irreversible statistical description, which although simpler, hecomes true

only as the pure time-dependent SchrZidinger state becomes so complicatedthatits

replacement by a mixed-state density matrix is justified.

Obviously this viewpoint sets forth an ambitious programfornuclearheavy-ion

physics. Rut we believe that it is justified by the fact that among all labora-

tory controlled events, nuclear heavy ion collisions are uniquely those in which

two systems initially at temperature zero (but far from equilibrium because of

their kinetic energy of relative motion) evolve in real time into an excited

state with a temperature of one or more MeVs. Moreover, these collisions are

also the unique laboratory examples of nuclear matter in flow. Therefore, if we

cannot come to understand the mechanistic details of thermalization and of

Fermidynamical fluidics here, we have little prospect elsewhere of doing so.

Thus we suggest that the discrepant nucleonic flow of Fig. 3, anomolous

as it is from the statistical energetic point of view, should be viewed as a

challenge to a deeper analysis, which offers a re-vivified hope for deeper

fundamental progress.

AS an early Step in exploring this hope, we present in the remainder of this

paper some simple results from a one-dimensional Schriidinger modelofthe nucle-

15oc J.J. Griffin, W. Broniowski / Open questions : thermalization and flow

onic flow in a dinucleus. The study was undertaken to see whether this simple

model behaves in accordance with the assumptions underlying the statistical des-

criptions. Thus for the Random Walk approach, the model might exhibit the role

of final state densities in determining the time evolution; or for the Randrup

analysis, whether the several physical assumptions made therein are validated in

this one specific example. As we shall see, the Schrhdinger model substantiates

neither statistical model's description of the early stages of the dinuclear

flow. It thereby underlines the need to study such models more skeptically and

in more detail.

4. DOUBLE-WELL MODEL OF 1D FLOW

A one-dimensional two-square-well independent particle modellO- of the

heavy-ion dinucleus is analyzed by obtaining its exact time-dependent solutions

numerically, and computing of the number, NR(t), of nucleons in the right well,

and its dispersion, uR(t). Then one inquires which features of the model system

determine their time dependence.

Figure 4 depicts the Double-Well model potential, V(x), and the initial

conditions for the time-dependent Schradinger equation,

1 (Ti+Vi)Y(?,t) = iH i(x',t) . i

The size parameters LR, LL and the potential difference, Vg, fix the double-

well potential, while the initial numbers of particles in the left and right

wave functions, NR, NL, together with the energy per nucleon in relative motion,

E/A, define the two ground state initial conditions.

We note that in Fig. 4, the difference between the right and left side Fermi

energies corresponds to the difference in the nucleon separation energies of a

more realistic finite well depth model, and to the energy difference between

adjacent integer N,Z points on its dinuclear energy surface. For the case

portrayed in Fig. 4, this driving force is towards the right; nevertheless, as

we shall see, the calculated drift is towards the left.

J.J. Griffin, W. Broniowski / Open questions : thermalization and flow 151c

5. DEPENDENCE OF 02(t) UPON CENTRAL DENSITY AND KINETIC PRESSURE

When the volume of the nuclidic potentials is given by the simplest prescrip-

tion, L=cN, one finds that the time dependence of 02(t) varies significantly

with N. This prescription specifies Fermi kinetic energies, EF, and overall

nuclear densities, N/L, which are independent of N. On the other hand, the

requirement that the average interior density be constant13-l6 leads to the

definition L = C(N+l/P) , (2)

and yields results for u* vs. time which are essentially independent of N for

N>2. Rut if one specifies the box volume so that the total kinetic pressure on

the wall, given by

PK = -(aKTOTAL)laL 3 (3)

is kept constant independent of N, there follows the alternative prescription

for the volume,

LR = CNR[l + 3/(2NR) + l,(2N;)]"3 . (4)

For this case, Fig. 5 shows that even the N=l dependence of u* is in close

agreement with the other N-values, and indicates thereby that the kinetic

pressure (3) is the determinant of the nucleon flow in this problem. We note

that the ratio of LR in (4) to its value in (2) approaches 1.0 as (0.961, 0.986,

0.993, 0.996,...) for (N=1,2,3,4,...), respectively. Thus, except for N=l, the

condition (4) of constant kinetic pressure (3) is approximately equivalent to

the condition (2) of constant average interior density. Unless otherwise

indicated, we use the volume condition (4) in all calculations reported here.

The results of Fig. 5 suggest that the nucleon fluxes for double wells

prescribed to have constant kinetic pressure are independent of N, and therefore

that zero drift in the average number of particles on, say, the right side,

should occur between such systems of different N (when VO=O and E/A=O). This

implication was also verified by explicit calculations not exhibited here.

One should notice also in Fig. 5 the effect of reflection from the distant

potential wall in limiting the size of o*(t). The times for a classical

152c J.J. Griffin, W. Broniowski / Open questions : thermalization and flow

particle moving with the Fermi velocity to reflect off the far wall and return

to the interface are indicated by the arrOws on the time axis. We believe that

the strong coherence of cur simple model for such reflections would not occur in

a more realistic three-dimensional model; therefore we restrict our inferences

about the flow to times shorter than these reflection times.

6. STRONG DEPENDENCE ON RELATIVE KINETIC ENERGY

The Double Well calculations exhibit the rather strong dependence of i2 on

the translational energy of relative motion, E/A, which is reported in Ref. 11,

but omitted here to save space. We note only that this aspect of the flow,which

is relevant to the energy dependence of the reaction,deserves further study.

2

"b'

7.

0.5-

04 -

0.3 -

0.2 -

0.1 -

’ ’ ’ ’ B A’ 1 J 0.0 0.2 0.4 I? 06 b.6 1.0 t

Tome I lP1cc)

FIGURE 5

0.3 ‘Vg 32 .O

WEAK DEPENDENCE ON POTENTIAL DIFFERENCE, VO

24--- 04 0.6 aa 1.0

Tmm (16%ec)

FIGURE 6

Figure 6 shows how s depends upon the difference, VO, between the potential

energy for a nucleon in the left and right wells. One finds that even potential

differences of 32 MeV, somewhat larger than the Fermi kinetic energy itself,

alter the slope, s, hardly at all whereas a much smaller translational kinetic

energy E/A=3.0 MeV had in Ref. 11 a noticeable effect. .

Also, we have found that the drift rate, d=NR, induced by such a 32 MeV per

nucleon potential difference is of roughly the same magnitude as the rate

induced by the kinetic pressure difference corresponding to an increase in the

average kinetic energy of only 8 MeV per nucleon. One concludes that a given

J.J. Griffin, W. Broniowski / Open questions : thermalization and flow 153c

increase in the kinetic energy per nucleon in, say, the left nucleus is rmch

more influential on the nucleon drift rate than the same increase in the

potential energy per nucleon. It follows that any description in which the

drift is proportional to a driving force which depends only on the sum of

kinetic and potential energies Nst contradict the present model.

8. FINITE DRIFT FOUND FOR ZERO CHEMICAL POTENTIAL DRIVING FORCE

In a simulation of the behavior of the flow of protons between a neutron

excessive nucleus and an N=Z nucleus, which is reportedin some details below, we

have formulated an example which explicitly illustrates this predominance of

kinetic pressure over potential energy in determining the flow. The example

consists of a series of Double Well calculations, starting at VO=O with perfec- .

tly balanced flows (i.e., such that NR=O). Then the potential of the left well

is increased by Vg, but its size is also increased so that the chemical poten-

tial on the left (equal to the total energy of the last particle, X=KF+VO) is

identical to the value it had at VO=O. Thus the total energy (kinetic plus po-

tential) of the last particle on the left remains for every value of VO exactly

the same as for the VO=O, NR=O case. Also, the right well remains unchanged for

all the calculations of the series. Then if the drift were determined by the

difference between the left and right chemical potentials, it should remain at

its VO=O value of zero for all of the (VO,LL) combinations calculated in Fig. 7.

We shall see that in fact the drift does not remain zero, but increases with the

increasing difference between the right and left kinetic pressures.

9. SIMULATED PROTONS DRIFT INTO AN N-EXCESSIVE NUCLEUS IN CONTRADICTION TO STATISTICAL DRIVING FORCES

Figure 7 exhibits the implications of the two-well model simulation of

"proton" flow between a neutron-excessive and an N=Z nucleus whose last proton

separation energies are equal. The precise potential for this simulation is

depicted in Fig. 4. There the Coulomb energy difference between the (right) Z=2

nucleus (assumed to have N=Z) and the (left) Z=3 nucleus (assumed to be neutron

154c J.J. Griffin, W. Broniowski / Open questions : thermalization and flow

excessive, N>Z) is simulated by the difference, VO, between the depths of the

right and left potential wells. Increasing VO therefore represents an

increasing proton charge; the number of protons does not change. At the same

time, for each value of VO the left volume is increased so that the Fermi energy

(and therefore the separation energy)

FIGURE 7 FIGURE 8

of the last proton, (E: = K:+VL) has precisely the same value as it had for the

VO=O case. This increased volume is meant to simulate the increase in A which

occurs as the neutron excess grows.

In this way we model the interfacing protons of a neutron excessive and an

N=Z nucleus whose last protons have nearly the same separation energy. Since

the separation energy of a nucleon in the left well is slightly less than that

in the right, the statistical "driving force" is small, but directed towards the

right, so that current statistical theories predict a proton drift towards the

right. Moreover, the ground state equilibrium points indicated as arrows on the

ordinate scale in Fig. 8 are also slightlytowardsthe right (i.e., tending to

increase NR), for VO=O.O, 2.0, and 4.0 MeV and slightly twards the left for

VO=8.0 and 16.0 MeV.

However, the Schrzdinger model shows for all VO>O values (in Fig. 7) a net

drift of "protons" into the (left) neutron excessive nucleus, which flow is

J.J. Griffin, W. Broniowski / Open questions thermalization andjlow 155c

either away from the ground state equilibrium value of NR (for VOz2.0, 4.0 MeV),

or far overshoots it (for VO'8.0, 16.0 MeV) during the early phase of the flow,

and which always contradicts the flow indicated by the higher chemical potential

of protons on the left. This flow results from the greater kinetic pressure,

PK, of the protons in the (right) N=Z nucleus and occurs here in spite of the

fact the energetic "driving force", a<E>/aNR, is oppositely directed. Since

this direction of the driving force is sufficient to guarantee a rightward drift

in statistical descriptions2-4 of this process, the present calculations

contradict these models.

10. EXTRAPOLATION TO SPHERICAL NUCLEI: LOCI OF KINETIC EQUILIBRIA

To obtain a first estimate of the Kinetic Pressures for spherical three-

dimensional nuclei, we calculate the total kinetic energy of a Fermi gas of

N-neutrons contained in a spherical box of radius RN,

KTOTAL = CI($$(N5'3) ,

where CI=(3/5)(%/2) 213

=3.5080. Then the kinetic pressure of this gas is given

by the derivative of KTOTAL with respect to the nuclear volume; one finds

PN = -a(KTOTAL)/a($ nR3) = C2(E ,,$[N5'31 3

where c2=(3/101r)(%/2) 213

= 0.55832. Analogous expressions apply the kinetic

pressure, PZ of protons in a box of radius, RZ.

We remark that the kinetic pressure varies inversely with the fifth power of

the radius of the neutron (or proton) distribution. Such sensitivity to radii

suggests that kinetic pressures will be strongly affected by differences among

the various nuclear radii (neutron, RN, proton, RZ, and mass, RA, respectively).

They may therefore provide a useful probe for their study. Conversely, the

detailed behavior of these radii may enrich the possibilities for flow patterns

in heavy-ion collisions.

156~ J.J. Griffin, W. Broniowski / Open questions : thermalization and jlow

In the simplest drop model, one has RN=RZ=rOA 113

for both target and projec-

tile nuclei. Then the difference in the neutron kinetic pressures of the

target-like (T) and projectile-like (P) nuclei is given simply by

p;-p; = c,[I(ql NT 513 _ (+5/q] , (7)

where CS=C2(,#/2Mr~). (Again, the analogous expression, with Z replacing N,

gives the proton kinetic pressure difference.) This

that for protons, both vanish along the same line

Zp/Np = ZTINT = ZC/NC

(Such a line, with slope 0.726 for Fe+Ho, and marked "Kinetic Equilibria" is in-

pressure difference, and

(8)

dicated in Fig. 1.) Here the subscript C denotes the combined dinuclear system.

Thus, the kinetic pressures are equilibrated along a whole continuu$of

dinuclear systems, in contrast with the energy equilibria, which occur only at

discrete points. Then were the kinetic pressures the sole determinant of

the flow, one would expect the system to drift towards this equilibrium line

and, having arrived there, to cease drifting altogether.

One is thus led to consider the nuclear flow to occur in two stages: (a) an

early rapid flow driven by the kinetic pressures, followed by (b) a slower

potential-driven flow along the locus of

improved generalization.

11. EQUILIBRIA CHANNELLEO (N,Z) FLOW?

Before closing, we wish to outline an

trates the possibilities of such kinetic

that of the kinetic equilibrium loci are

kinetic equilibria given by (8), or its

interesting speculation which illus-

dominated flows. We have noted above

one-dimensional continua instead of

isolated points. Moreover, there seems to be no reason why in general the locus

of neutron and proton equilibria should coincide, as they do in the special case

characterized by Eqs. (7) and (8). Then suppose that due to differences in RN,

RZ (such, for example, as those described by the droplet model17) the single

locus (8) splits into two separate loci of equilibrium, one for the neutron flow

and one for protons, as indicated in Fig. 8.

In this case if the system finds itself between these lo&, it could be

forced to follow what might be called a "Kinetic ~quiljbrium Channelfed Flow" on

the (W,Zi plane. One need anfy note that inside this ~uilibrium channel the

tendency would be to flow vertically downwards from locations near the neutron

locus and horizontally to the left from those near the proton locus, resulting

in a drift towards lighter projectile-like fragments. We emphas$ze that the

djrection of this constrained flow is determined solely by the relative

positions of the two loci. If the loci were jnter~hanged~ the flow would be

guided vertically upwardsandhorizontally to the right along a line of growing

mass towards the symmetric configuration (at which the two equilibrium loci must

cross). All physically realizable flows rmst, of course, conform to the

constraints imposed by conservation of energy. Still kinetic channelling might

impose its own tendencies, which reflect the kinetic energies rather than the

total energies of the nuclei, upon the tendencies of the total energy surface,

providing thereby some perhaps interesting alterations of its predictions.

Obviously, one wonders whether the case of a7Ca+20gBi mentioned above* might be

a candidate for such interpretation.

The major result to emerge from the Double-Well model is that the kinetic

energy of the nucleans, as quantified by their kinetic pressure (3), is far more

important than their potential energy in "influencing the early nucleonic flow.

This result, in turn, is sufficient to guarantee contradictions between this

Schrb'dinger model flow and that described by the statistical models; as follows,

Since the statistical amdefs generate "driv‘ing forces" on the basis of total

(potential plus kinetic) energy, tlley predict (e.g.) that two different proton

distributions which have equal proton separation energies will feel no statis-

tical driving force and suffer therefore no proton drift. In the SchrBdinger

158~ J.J, Griffin, W. Broniowski / Open questions : thermalization and flow

Uouble-Well model, even when the proton separation energies are nearly equal (as

in the calculations of Fig. 7), their kinetic pressures may be very unequal

according to Eq. (7), if only their average proton densities differ. Then

Double-Well will predict a net proton flow where the statistical description

predicts none.

Conversely, the Double-Well calculations show that when the kinetic pressures

of two distributions are equal, even very substantial differences in potential

energy do not generate large net nucleonic flows. Thus the Schr6dinger calcula-

tion will in some cases predict small drifts for target projectile combinations

which, from the statistical dinuclear energy viewpoint, involve very strong

driving forces.

A second qualitative result of the present study is that the early flow of

nucleons it describes bears no evident systematic tendency to mOve even in the

general direction of the ground state equilibrium (N,Z) configuration. It

therefore suggests that the whole statistical method of analysis in terms of

small deviations from equilibrium may be misleading for these systems which are

initially so far from equilibrum. Instead the early stage dynamical evolution

of the system seems to depend not primarily upon the properties of that special

ground state equilibrium point, but instead upon the specific properties of its

own present state. Clearly, a validation of this general viewpoint would

significantly expand the possible complexity of the heavy ion process, opening

many interesting possibilities which would be foreclosed in a system whose flow

is dominated by the immediate need to move towards equilibrium.

Finally, one is led to recognize the possibility that the Fe~idynamic

fluids exhibits an early, rapid, kinetic driven flow and a subsequent, slower,

potential driven flow. Also, that specific details of the separate neutron and

proton volumes might lead to drift patterns constrained by the continuous loci

of the separate neutron and proton kinetic pressure equilibria to follow paths

quite at variance with the implications of the overall energetics which our

statistical descriptions have so far been based on.

J.J. Griffin. W. Broniowski / Open questions : thermalization and flow 159c

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