Nuclear hydrodynamics for ellipsoidal shapes

Nuclear Not to

Physics

A341 (1980) 513 - 532;

@ North-Holland

Publishing

Co., Amsterdam

be reproduced by photoprint or microfilm without written permission from the publisher

NUCLEAR

HYDRODYNAMICS

FOR ELLIPSOIDAL

SHAPES +

R. L. HATCH W. K. Kellogg Radiation

Laboratory,

California Institute

of Technology,

Pasadena,

California 91125, USA

Pasadena,

California 91125. USA

and A. J. SlERK ++ W. K. Kellogg Radiation Laboratory,

California Institute

of Technology,

and Theoretical

Division, Los Alamos Scienrific Laboratory, University New Mexico 87545, USA t

of California,

Los Alamos,

Received 27 December 1979 Abstract: The liquid-drop moue1 is used to study the dynamics of nuclear matter confined to an ellipsoidal shape of constant volume. The equations of motion for the two degrees of freedom of the shape are solved for irrotational, incompressible hydrodynamic fluid flow with hydrodynamic shear viscosity as the means of energy dissipation. Angular-momentum effects are approximated by means of a centrifugal pseudopotential. Non-axially symmetric shapes are found to be crucial in the intermediate stages of a nuclear collision, where elongation of the compound nucleus may occur either at right angles to or along the original line of centers, depending on the collision energy and the amount of dissipation. Scattering of identical heavy nuclei is considered in a simple model with qualitative and approximate quantitative predictions made for deflection functions and differential cross sections at low energies. A new inelastic peak in the differential cross section is found which is dependent on the hydrodynamic assumption and may provide an experimental observable which can distinguish between short- and long-mean-free-path models of nuclear matter flow. Equilibrium configurations for the ellipsoid are also presented and a comparison is made to the results of more general parametrizations.

1. Introduction

Macroscopic models have been used extensively for several years to study heavyion reactions ’ -“). There is a great computational advantage for large systems over microscopic theories, which must treat many internal degrees of freedom. One commonly used macroscopic theory incorporates incompressible, hydrodynamic fluid flow to model the flow of the nuclear fluid in dynamic nuclear processes 536). Nuclear dissipation, the transfer of energy from collective motion to internal degrees of freedom, is often included by letting the fluid have a finite shear viscosity “). This ’ Supported in part by the National Science Foundation [PHY76-836851 and in part by the US Department of Energy [W-7405-ENG-361. ” Alfred P. Sloan Foundation Research Fellow 1975-1977. * Present address. 513

514

R. L. Hatch and A. J. Sierk 1 Nuclear hydrodynamics

assumes that dissipation occurs by individual two-particle collisions and is called two-body dissipation, as opposed to one-body dissipation which results from singlenucleon collisions with a moving potential wall ‘-lo) . Calculations in the framework of the dynamic liquid-drop model have shown that both types of dissipation are able to account quantitatively for observed most probable fission-fragment kinetic energies throughout the periodic table 6*9- ’ ‘). Recent time-dependent Hartree-Fock calculations 12) do not exhibit the incompressible hydrodynamic flow seen in the liquid-drop calculations. It is important that the consequences and predictions of different theories be better known so that it is possible to distinguish among them. It is with this in mind that we make the present investigation of nuclear dynamics using incompressible, hydrodynamic flow with shear viscosity. In this study, axially asymmetric shapes are allowed by using an ellipsoidal shape parametrization. Sect. 2 describes the derivation, in the framework of the liquid-drop model, of the equations of motion governing the time developement of the shape. In sect. 3 the equilibrium configurations for a rotating ellipsoid are compared with those for more general shape parametrizations. Dynamic effects are investigated in sect. 4 with qualitative and approximate quantitative predictions given for inelastic heavy-ion scattering. Finally, in sect. 5 we summarize our results.

2. Theory We derive the dynamic equations of motion from the generalized Lagrange equations. As in all calculations of this type we desire a parametrization which is general enough to allow freedom of shape evolution without requiring excessive computation. In the simplest model allowing the axially asymmetric shapes desired, the nuclear matter is constrained to an arbitrary ellipsoidal shape x2 2 -+y+-=l, a2 b2

22 c2

where a, b, c are the semi-axis lengths. We assume that the nuclear fluid is incompressible, so that the nucleus maintains a constant volume

where R, = r,A* is the radius of the sphere with a = b = c. With this constraint we need treat only two of the three axis lengths as dynamic variables. These ellipsoidal shapes represent quite accurately equilibrium shapes which are not greatly deformed from spherical 13). The major obvious drawback of this shape parametrization is the lack of any neck formation, which becomes a significant restriction at larger deformations and which prevents fission from occurring. We assume that movement of the nuclear fluid during deformation is irrotational, hydrodynamic flow. A comparison with the exact viscous flow for small amplitude

515

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

motions about a sphere is given in the appendix. For motion confined to an ellipsoid, the fluid velocity vector is 14) l&X,

y, z) = ;

xt+$ yf+ f zii.

Here ci is the time rate of change of a, etc., and i,j? and E are unit vectors along the Cartesian x-, y- and z-axes, respectively. By assuming that the nucleus rotates as a rigid body about the c-axis with angular velocity o, the effects of angular momentum can be approximated; thus 15) Uror = VDEF - wy;+ ox;.

(4)

The total kinetic energy is r6) T = + pmuTOT.u =oTdI/ = +2+6’+C2)+

$‘(a2+b2),

(5)

s

where the constant mass density pm is M/I/. Nuclear dissipation is assumed to be of the two-body type, and is included as ordinary hydrodynamic viscosity. For irrotational flow the rate of energy dissipation is i’ ) dEdis - p V2(v,,,. dt s

u&dV

= 2pV

d2 f$ + p + $

>

,

where p is the coefficient of shear viscosity. We define the potential energy according to the usual liquid-drop for a deformed nucleus: I’ = E s-E’“‘+&-E~‘. s E, is the nuclear surface energy and is simply proportional Coulomb energy EC is given by

(6) prescription

(7) to the surface area. The

(8) E(sO)and Eg’ are the corresponding

energies for a spherical shape. We write them as

(9)

In our calculation we use the values r. = 1.2249 fm, a, = 17.9439 MeV, and the surface-asymmetry constant K = 1.7826 [ref. ‘“)I. For an ellipsoid the necessary energies are expressed in the following relations 19)

516

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

(with c < a c b):

k2

C

k2 s

_

=

b’-”

(10)

b2-_

’

b2(a2 - c2) a2(b2 - c2) ’

cp = arcsin [(b’ - c2)+/b]. F(cp, k) and E(cp, k) are the incomplete elliptic integrals of the first and second kind defined by r’p (1 -k* sin2 8)-+dO,

F(cp, k) =

(11)

We evaluate them using the process of the arithmetic-geometric descending Landen transformation 20). By defining the fissility parameter

mean and the

(12) we rewrite eq. (7) as I’ = {(B,-

1)+2x(Bc-

l)}E’,“.

(13)

To introduce the frictional forces in the Lagrange formalism, we use the Rayleigh dissipation function 21)

F=!dEdis 2 dt

(14)

’

The classical equations of motion are then given by d aL aL ---=--9 & () aqi agi

aF

i=

1,2,...N,

(15)

a4

where L = T - V is the Lagrangian, T is the kinetic energy, and I/ is the potential energy. In our case (N = 3), ql, q2, and q3 can be chosen to be two of the three semi-

R. L. Hatch and A. J. Sierk 1 Nuclear hydrodynamics

517

axis lengths (a and b for example) and the angle of rotation cp (ci, = w). This gives three coupled, second-order non-linear differential equations. Since cp is a cyclic coordinate, its Lagrange equation yields the conservation of angular momentum and can be reduced to first order: $%$a2 + b2) = 1.

(16)

It is also convenient to define two other dimensionless quantities (17) y is the rotational energy of a rigid sphere with angular momentum 1 in units of the surface energy Eke’ [ref. “‘)I. z is the viscosity in natural units. Values for x, y and z define completely the physical properties of the system and together with the initial conditions for the shape and motion of the ellipsoid, they specify uniquely the time development of the system. The equations of motion are solved by expressing them as a system of five coupled, first-order, non-linear differential equations and then integrating numerically using a fourth-order Adams-Moulton predictor-corrector method 23), with the starting procedure based on a modified fourth-order Runge-Kutta method 23P24).A check of the numerical accuracy is made by simultaneously integrating the rate of energy dissipation and comparing with the total energy of the system. In a typical dynamic process representing the intermediate stages of a collision (from coalescence to contin.ued elongation as pictured in fig. 3), the energy is conserved to about one part in 106.

3. Equilibrium shapes To begin to understand the dynamic processes, it is necessary to know the features of the effective potential energy surface I/

eff

(a

9

b) =

~/(a

3

b)+

!!!?8!! a2+b2

’

(18)

We have here let the two independent variables be a and b (c is the axis of rotation). Of special importance is the location of minima, maxima and saddle points and how they correspond to known shapes for less restricted parametrizations, i.e., those that allow formation of a neck structure. With this information we can estimate the regions of validity of the ellipsoidal parametrization. Equilibrium configurations for ellipsoids have been previously investigated in various approximations 13,25,26). In ref. 25) ellipsoids with an axis of symmetry (spheroids) were used to describe equilibrium shapes of nuclei. Triaxial equilibrium shapes were treated in ref. i3) by an expansion of the Coulomb and surface energies

518

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

valid for small asymmetric deformations of an ellipsoid about a symmetry axis, and a sophisticated expansion was used in ref. 26) to accurately calculate shapes of minimum energy for asymmetric ellipsoids. We find these points of equilibrium by solving for sets of values of a and b which satisfy

d&f -_ = 0 and TF da

= 0.

(19)

This was done using the Newton-Raphson method 27) of iterative solution. A point where the effective potential is a minimum corresponds to a stable equilibrium shape whereas saddle points and maxima correspond to equilibrium shapes of one and two degrees of instability, respectively. We make no distinction between instability and secular instability. For a detailed description of the rotating-liquid-drop-model equilibrium configurations we refer the reader to ref. 22). We will touch only lightly on the main features of this work and their relation to the ellipsoidal parametrization. For y = 0 and x < 1 the sphere is a stable equilibrium (ground state) shape. As y increases, the sphere flattens at the poles as the other two axes lengthen while remaining equal to each other. When y reaches yi(x), the bifurcation point, this oblate equilibrium shape becomes unstable. For x < xc where xc z 0.81, a triaxial stable equilibrium shape forms at yr(x) and continues to exist up to a point y2(x) where equilibrium is lost altogether. Fig. 1 presents the axes of the equilibrium shapes for an ellipsoid with axes Rmin 5 %ed S &,,a, (&in is the axis of rotation in each case) and for x = 0.0, 0.3 and 0.6. The upper curves correspond to the oblate equilibrium shapes with the points of beginning instability (yr) marked by the dots along these curves. The lower curves which branch off at this point are the stable triaxial configurations. The lower dots mark the points y, [taken from ref. ‘“)I where serious discrepancies of the ellipsoidal model arise for x < xc. For an ellipsoid, unable to form a neck, the stable triaxial equilibrium shapes continue to exist indefinitely for larger values of y (now marked by the dashed curve to signify that it is spurious). In contrast to this, the lines of exact equilibrium shapes curve downward and to the left at y, and continue back to reach y = 0 with reduced values of Rmin and Rmed. These are saddle points and therefore unstable. For y = 0, their properties are well known from previous studies [refs. 5, 2E,““)I. That such discrepancies occur for an ellipsoid is to be expected for such large deformations where neck structure becomes important, for in the absence of a neck the surface energy contribution to the effective potential increases without bound as the deformation increases. As x increases toward xc, y, approaches yr. Thus the triaxial stable equilibrium shapes cease to exist for x > xc and the equilibrium curve for the unstable triaxial saddle-point shapes joins the oblate-spheroid curve at y,. This can be seen for x = 0.9 in fig. 2, which gives the ellipsoidal equilibrium curves for x = 0.8, 0.85, and 0.9. These triaxial saddle-point shapes are reproduced entirely for x > 0.887

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

519

O.DLLL . . . . . RotatIonal

0.0

0.0

’

’ 0.2

’

1 0.4

Rotationa

Energy y

1

1 0.6

1

1 1 0.6 I.0

I Energy y

Fig. 1. The ratios of axes R,,,/R,,, and R,,,/R,_(R,,, 5 R,,,_, 5 R,,,, Rrninis the axis of rotation) of ellipsoidal equilibrium configurations for fissilities x = 0.0, 0.3, and 0.6 as functions of the rotation parameter y. The upper and lower dots along the curves correspond to values of y, and y,, respectively. y, is the critical rotation when stability against triaxial deformation is lost. The critical rotation y, [taken from ref. 22)] marks the loss of stability (and equilibrium) for true equilibrium shapes SO that the equilibrium curves beyond this point (marked by the dashed lines) are spurious.

(e.g., x = 0.9, see fig. 2), only partially for 0.887 > x > xc (e.g., only for y 2 0.016 for x = 0.85, see fig. 2), and, as we have already seen, not at all for x < x,. The spurious, stable equilibrium curves in fig. 2 are again denoted by a dashed line. Note that yr(x) -+ 0 as x + 1 so we recover the familiar result that all stability is lost for x > 1. This comparison indicates that an ellipsoidal parametrization reproduces most of the features of the potential-energy surface for shapes that are not too deformed from spherical, and therefore especially for large values of x where the equilibrium shapes do not show the neck structure of more deformed shapes. These features of the ellipsoidal approximation must be kept in mind when making dynamical calculations

520

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

so

that we do not draw wrong conclusions when the shape evolves into regions where the parametrization is inappropriate.

1 1 3 1 1 1 1 1 1 0.03 0.04 0.05 0.00 0.01 0.02

0.0

Rototlonol

Energy y

0.0

’ 1 1 1 1 1 1 1 0.00’ 0.01 0.02 0.03 0.04 0.05 Rototionol

Energy

y

Fig. 2. The same as fig. I with fissilities x = 0.8, 0.85, and 0.9. As x increases towards xc = 0.81, y2 approaches yl. (* The exact point for the loss of equilibrium along the curve for x = 0.8 is not accurately known.)

4. Dynamics The principal reason for studying the dynamical properties of the system is to extract information which will be useful in comparing this type of hydrodynamic flow to other theories and to experiment. We will now investigate several features of incompressible, irrotational, hydrodynamic flow in dynamic processes corresponding to nuclear collisions in order to define results characteristic of this model. We must also clarify several conventions used in this section ‘Since x, y and z describe the entire physical properties of the system, we will use them quite extensively along with the natural units such as energy in terms of E(O)and distance in units of

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

521

R,. However, many times for the sake of intuition or convenience we will also use the corresponding nuclear-physics values, e.g., A, Z, 1, p, E in MeV, etc. In doing this Z and A will always be found from x using Green’s approximation to the line of beta stability 30)

We will consider the viscosity p to be a general property of nuclear matter and the value used in this work, unless otherwise stated, p = 0.03 TP (1 terapoise = 10” dyn s/cm2) is one that gives good agreement with experimental fission-fragment kinetic-energy data il). All results will be given in the c.m.s. As two nuclei collide in the hydrodynamic model, the nuclear fluid in the plane between the nuclei and perpendicular to their line of centers is forced to the outside by the fluid pushing in from both ends, thus causing a change in direction of part of the collective motion of the system. The degree to which this “bulging” occurs is of course determined by the energy of the collision. In contrast to this is the flow pattern of one-body theories, where the nucleons pass by each other seeing only the potential “walls”. This lack of bulging is clearly seen in recent time-dependent Hartree-Fock calculations of I60 + I60 and 40Ca+40Ca reactions 12). One of the interesting ways the hydrodynamic bulging expresses itself is in the breakup of two colliding nuclei for y > y,. For energies below a critical value E,(y), the most compact configuration of the colliding system is such that the dimension of the compound nucleus along the original line-of-centers is greater than the dimensions in the plane perpendicular to it, and the system fissions along the original line-of-centers in the rotating frame. At larger energies the axis perpendicular to the axis of rotation and original line-of-centers becomes larger and fission then proceeds perpendicular to the original line-of-centers. In the remainder of this paper we refer to these processes respectively as line-of-centers and right-angle fission. This process is pictured in fig. 3 for energies just below and just above the critical energy E,. The existence of this phenomenon can also be inferred from general properties of l&&r, b) of eq. (18), where c is along the axis of rotation. For y > y, the unstable oblate equilibrium shape is a saddle point. Since I/,&, b) is symmetric in a and b, it falls off on each side of the ridge a = b. For energies near E,, the maximum compactness occurs close to this unstable equilibrium point but on a specific side of the ridge along a = b, with this determining the direction of fission. In calculating values of El we started the nucleus in a prolate shape a = c = ib with ci = C > 0 (as in fig. 3). Notice from figs. 1 and 2 that this distortion is in a region where the parametrization is expected to be approximately valid. Figs. 4-6 give the critical initial radial kinetic energy El versus y for various values of X. Fig. 5 also shows effects of viscosity with the curves labeled by values of viscosity.

522

TIME Units of EZ.085 Et:‘(57Mav)

Q 8

0.0

0.4

1.0

2.0

4.0

Q CJ 8

6.0

0

8.0

x ~0.85 (A=263.2=102)

‘a

y=O.l89 (l=l98n 1 /L~O.O~TP

Fig. 3. The intermediate stages of a collision as seen perpendicular to the scattering piane for energies just below and just above the critical energy E, = 0.084@‘, showing the transition from fission along the initial line-of-centers (denqted by the bisecting line) to fission along a perpendicular direction at slightly larger energies. The given energies are the initial radial kinetic energies of the prolate shape with axis lengths n = c = fb at time r = 0.

Right-angle fission does not continue for all energies, but only until the colliding nuclei slam together with enough force that they “spring” back along the initial lineof-centers. Right-angle fission disappears for these very energetic collisions because the triaxial instability has no time to develop before the system rebounds along its initial direction ofmotion. These highly energetic collisions will not be treated further

/

R. L. Hatch and A. J. Sierk

Angular 050

100

Nuclear

Momentum 150

1 (units of h)

200

I

250

I

x=0.3(A=75,2=33)

$0

Lb ~

523

hydrodynamics

210

nr\

3!0

Ratatlanal

410 Energy

5:0

7 E!

6.b

y

Fig. 4. The critical energy E, for the transition between line-of-centers and right-angle fission (as pictured in fig. 3) as a function of angular momentum for fissility .Y = 0.3. The energies are the initial radial kinetic energies of the prolate shape with axls lengths a = c = fb.

Angular Momentum 0 50 100 150 I I I

I 200 I

(units

of h) 250 -‘O.Et

x=0_6(A=l73,2=70)

+lz 0.6 $ ? c 0.4 ; z 0.2 u

0.0

0.2

0.4

Rotational

0.6 Energy

0.0

0.0

y

Fig. 5. The same as fig. 4 but showing the effect of viscosity on the system x = 0.6.

Angular Momentum I[unlts of h) i(J 100 150 200 250

I

a12

I

1

1

0.3

x=.65(A=263,2=102)

^ ow 0.00 z In z z- 0.04

0.00

-F 0.2;

c > % 0.1 2

,0.0

I

0.1 RotatIonal

I

I

0.2 Energy

0.3 y

Fig. 6. The same as fig. 4 for x = 0.85.

0.0

524

R. L. Hatch and A. J. Sierk 1 Nuclear hydrodynamics

in this paper, but we do compare these new critical energies E, with E, for x = 0.6 in fig. 7. Angular 050100 I.01 I

Momentum 1 200 150 1

Rototlonol

I

(units of h) 250

13.0

Energy y

Fig. 7. Comparison of the critical energies E, and E, for x = 0.6. E, is the energy at which line-ofcenters fission agam occurs, thus dividing the graph into two regions: the region between the curves where right-angle fission takes place and the region above the curve E, and below the curve E, where fission is along the initial line-of-centers.

We now estimate the scattering angles 8(B) (B = impact parameter) for heavy-ion collisions with a simple model. The spherical nuclei are assumed to be deflected by a pure Coulomb potential until contact is made. At this point they are arbitrarily given a prolate spheroidal shape of Rmin = Rmrd = &,,,,, &,in = Amed> 0 with the initial kinetic energy found by conservation of energy. Orbital angular momentum is assumed to be conserved throughout the entire process. The time evolution of the system is followed until fusion is seen to occur or until the shape again elongates to Rmin z Rmed M ‘R 2 m3xwhere scission is assumed to take place. Using the dissipated energy to calculate the final fragment kinetic energy, the final trajectory is again a Coulomb orbit. These relatively crude approximations are necessary to preserve the simplicity of our model. The quantitative details of our results are thus not to be taken too seriously. However, the qualitative results are inescapable, and point to an interesting phenomenon that will occur only in a model with hydrodynamic-type flow. Due to the hindrance of a neck structure for an ellipsoid and the arbitrariness of the initial and final shapes, the scattering angles calculated might be off by several degrees, though the error is decreased by the fact that in the initial and final stages of the process the shape changes rapidly with a small angle of rotation (see frg. 3). We do expect, however, that this error changes fairly slowly with the impact parameter B, so that the general features of the scattering will be preserved. Fig. 8 shows such a calculation for the system x = 0.85 with an initial kinetic energy at infinity of 1.8 MeV/nucleon. (The interaction barrier for this system is 1.13 MeV/nucleon.)

R. L. Hatch and A. J. Sierk / Nuclear ~~d~od~na~ics

0.0

Impact

Parameter

0.2

0.4

525

(umts of Ro)

0.6

0.8

1.0

1.2

-1601

Fig. 8. The deflection function B(B) for the collision of identical liquid drops with initial c.m. kinetic energy E = 0.7 Ez”) and combined-system fissility x = 0.85 showing partial orbiting at two impact parameters and a rainbow in between. These general features appear over a wide range of energies and system sizes. The discontinuity at B z 0.96 R, is due to the incorrect treatment of the neck during fission in the ellipsoidal approximation.

The corres~nding (unsymmetrized differential cross section shown in fig. 9 was numerically calculated with the classical formula -W@ g----BAB dQ d(cos 0) ’

II

I*

‘1,

t

io.o-

,I

1

I

x =.85(A=263,2=102)

I

i t

7

I

(21)

6.0

E=071 . E~“(473~V) l p=O.O3 TP

I

I

,

5.0

i

L

I I \

40-

. Jqz -

1

3.0

,~~~~:__

2.0

s b

1.0

to

20 30 CM Scattering

40 50 60 Angle (degrees)

7

Fig. 9. A comparison of the inelastic and elastic parts of the differential cross section calculated from the deflection function of fig. 8. The discrete points were calculated from eq. (21). The inelastic peak at e Elm.s 18 degrees is a general feature over a wide range of energies and system sizes. The small peak at 35 degrees is a poor approximation to the Coulomb rainbow, due to the discontinuous treatment of the neck during fission (see fig. 8).

526

R. L. Hatch and A. .I. Sierk / Nuclear hydrodynamics

using A0 = 2O. We see that the most interesting areas of the deflection function, i.e., the areas which contribute most to do/dQ are those for which the slope is small and slowly varying; e.g., at the maximum a,,, near B = 0.37. For larger values of B, the curve becomes progressively steeper until the point B z 0.96 where pure Rutherford scattering begins. The discontinuity in this area is due to the discontinuous way scission is forced to occur from an ellipsoidal shape and does not indicate any physical effect. The general features of fig. 8 are not spurious results of the ellipsoidal shape parametrization but can be explained from the basic principles of the potential energy surface. For B 5 0.2, where y is less than y,, a stable, oblate spheroidal equilibrium shape exists. The value of viscosity is large enough that a collision is quickly damped, preventing the attainment of the fission barrier, and the nuclei fuse together in a classically stable shape. Stability is lost gradually as y increases through y, so that fission first occurs slowly near B = 0.21, resulting in large negative angle scattering there. The process at these small impact parameters is that of right-angle fission since the radial energy of the collision is greater than E,. As the impact parameter increases with more energy being rotational, the radial kinetic energy decreases through E, resulting in the shape passing through an unstable equilibrium point and the transition to line-of-centers fission. Large negativeangle scattering again takes place as the shape “sits” near this unstable equilibrium point causing the steep downward curve below B = 0.87 and the even steeper rise (due to the ninety degree upward displacement at transition) above B = 0.87. The existence of the “interesting” maximum between the two sharp downward peaks (at B z 0.21 and B cz 0.87) follows automatically. These general features are seen in the entire range of calculations we have made (from x = 0.6 to x = 0.99 with p = 0.03 TP), b u t only for energies below E,. (Above E, more complicated motion is found.) For lighter systems than x = 0.6, where viscous effects are not as strong (see appendix), more energetic collisions may reach the fission barrier for y < y, again causing more complicated behavior. Other important scattering information is the final kinetic energy of the nuclei. Because of the uncertainties in calculating this quantity, we can only state qualitatively that the final kinetic energy increases monotonically with B and only very slightly for events with angles near I$,,,, for large changes in the incident kinetic energy. Since the main observable feature is the maximum I!&,, a more thorough discussion of its properties is in order. That the maximum occurs at such a small scattering angle presents a problem as can be seen from fig. 9, which compares the Rutherford scattering with the inelastic part of the scattering. It should again be mentioned that the exact scattering angle may not be extremely accurate (the position of 8,,, may be shifted), but what can best be investigated here is how 8,,, varies with incident energy and with the properties of the nuclei themselves. As the incident energy increases, the processes speed up and fission occurs earlier. Thus the value of the peak maximum t9,,, increases. We give e,,,,, as a function of incident energy for x = 0.85 in fig. 10. As can be seen, going to energies higher than

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

Coulomb

1.0

1.4 E;“;

521

Barrier

1.8 W/2n;2cleon)

2.6

3.0

Fig. 10. The angular position of the inelastic peak in the differential cross section (see fig. 9) as a function of incident c.m. kinetic energy for the x = 0.85 system.

3 h&V/A will not increase 8,,, by a large amount. Also, at higher energies a larger proportion of the incident kinetic energy is dissipated in the collision so that in transforming from c.m. to laboratory coordinates the inelastically scattered fragments transform to more forward angles than do the elastically scattered nuclei. Bringing the inelastic scattering out of the forward elastic-scattering peak is therefore difficult for this system. By going to lighter systems, the nuclei scatter through a larger angle before contact is made. This explains 8,,, becoming more negative as seen in fig. 11 (with incident energy 1.5 MeV/nucleon). Since the exact value of viscosity is still uncertain, it is also interesting to see the effect of varying this parameter. The x = 0.85, E = 1.8 MeV/nucleon results are plotted in fig. 12. It is seen that the value of viscosity plays an important role in the quantitative analysis of scattering with the effect being as expected; the less viscous reactions proceed more quickly. This is yet another uncertainty in our calculation of the scattering angle even though the qualitative features remain unchanged. The sensitivity to dissipation raises the interesting possibility of determining the strength of viscosity by comparing experimental results to a model calculation with more shape freedom. We point out that the peak in the cross section at le,,,,,l stands out better when 8,,, is negative because the only contribution in the deflection function for l0l< ]B_( is from the steep curve connecting the orbiting with the Rutherford scattering. This

528

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamrcs 30

I

,

I

,

I

,

I

,

EynF = 1.5 MeV/nucleon

-301 0.5

’

’

0.6

’

’

0.7 Fwltty

’

I /

’

0.0 x

’

’

0.9

’

I

1.0

Fig. 1I. The angular position of the inelastic peak in the differential cross sectlon as a function of the system size at a constant cm. bombarding energy per nucleon of I .5 MeV.

is in contrast to the process pictured in figs. 8 and 9 for which tJm3,is positive and there is a large contribution for 10)< JB,,,I. As a consequence of the results pictured in fig. 11, we recommend that in searching for this peak one use lighter systems (such as x z 0.6) to maximize the chances for finding 8,,, negative. In summary, we have calculated the scattering of nuclei in a simple model. The deflection function B(B) is characterized (in the energy range considered) by two sharp downward peaks (partial orbiting) with a broad maximum peak (rainbow) in

Viscosity

p

(TP)

Fig. 12. The effect of viscosity on the angular position of the inelastic peak in the differential cross section for x = 0.85 and c.m. bombarding energy of 0.71 Go’.

R. L. Hatch and A. J. Sierk 1 Nuclear hydrodynamrcs

529

between. Even though the exact location of this peak is not known accurately, the general features are that it is shifted downward (toward more negative angles) by decreasing the incident energy, increasing the viscosity, and by going to lighter systems. This is experimentally interesting, since the peak also corresponds to a peak in the inelastic part of the differential cross section.

5. Summary

and conclusions

We have used an ellipsoidal parametrization to investigate effects of triaxial shapes on equilibrium configurations and dynamic processes in the liquid-drop model. The fluid flow was assumed to be incompressible, irrotational, hydrodynamic flow with the corresponding hydrodynamic viscosity included as a means of energy dissipation. Angular momentum was included by rigid-body rotation with a centrifugal pseudopotential. In sect. 3 we showed that the equilibrium configurations of an ellipsoidal shape are qualitatively the same as unconstrained configurations when deformations are not too large. At larger deformations an ellipsoid produces spurious stable equilibrium shapes and fails to produce triaxial saddle-point shapes for x < 0.81. In considering dynamical processes in sect. 4, the triaxial freedom was found to lead to interesting phenomena in processes corresponding to nuclear collisions. Collisions with energies above a critical value E,(y) were found to have their collective motion transferred from along the original line of centers to the direction perpendicular to it with fission occurring along this direction. The effects of this process on the deflection function 8(B) and its relation to a broad, characteristic peak in 8(B) were discussed. This characteristic peak corresponds to a maximum in the inelastic part of the differential cross section and although its position is not accurately known, 8,,, shifts toward more negative angles with decreasing incident energy, increasing viscosity, and decreasing size of the system. The inelastic peak was observed in the mass range x = 0.6 to x = 0.99 but is expected to be more easily observable for the lighter systems in this range. These results depend crucially on the hydrodynamic assumption and should not be seen in one-body models such as timedependent Hartree-Fock, thus providing an observable phenomenon which distinguishes between the two types of models. A defect of our approach is the impossibility of neck formation in the ellipsoidal shape. However, we expect that the qualitative features of the processes considered in this paper will not be altered by a more general parametrization, and we have tried throughout to explain these features using general properties of the system (such as arguments about the potential energy surface). A more accurate treatment will have to await a more general parametrization.

530

IRROTATIONAL-FLOW

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

APPROXIMATION

We investigate the use of irrotational flow as an approximation to exact viscous flow. The exact solution to the Navier-Stokes equation for an incompressible fluid has been solved for small distortions about a sphere with the boundary represented by r = &I + s(t) I;,(& cp),

(A.1)

where x,(6, cp) is a spherical harmonic 31,32). To each order in 1 there corresponds an infmite number of radial eigenfunction solutions for the fluid velocity vector. The time dependence can be written as s(t) = s,,e -szkf where k is the order of the radial eigenfunction. The two lowest values of (Tfor each I are complex-conjugate pairs for underdamped oscillations or are both real and positive for overdamped motion. For all higher k modes, G is both real and positive. For E < 1, an 1= 2, m = 0 distortion represents an axially symmetric ellipsoidal shape (spheroid). By solving eqs. (15) with a = c, b = R, +&e-O’, no rotation and the irrotational flow assumption, we find that CJis given by 5 ts -=00

+

a2 -

(25_/)? a4

64.2)

where (31 - x)ME’O’}+ a2 = _.Z.___L--, rV& and

0; = {$(1-x)} 2; 0

is the natural (frequency)= of an undamped oscillation. We compare this in fig. 13 with the lowest eigenmodes for the quadrupole oscillation in the exact solution. The expected agreement at large a is also seen analytically by solving eqs. (A.2) in this limit. This yields d -_= 00

5+i

a2 -

as

a+

co,

(A-3)

which is the same limit as the exact solution 32). The irrotational value is seen to be better for larger a. With p = 0.03 TP, a2 x 5.6 for X = 0.85 and a2 z 8.65 for X = 0.6. The first case is close to critical damping and represents an increase of about 41 “/, in the real part of rr, while the latter case is an increase of about 37 % over the exact value of Re (0). It should again be mentioned that this comparison is only valid for small oscilla-

R. L. Hatch and A. J. Sierk / Nuclear hydrodynamics

I

,

I

Quadrupole

,

I

,

531

I

Motion

Fig. 13. A comparison of the exact and irrotational values of the characteristic frequency c for small quadrupole oscillations about a sphere. The exact values are taken from ref. s*).

tions about a sphere and that less periodic motion, such as motion towards fission may result in the buildup of large vorticities resulting in even greater errors.

References 1) A. J. Sierk and J. R. Nix, Proc. of the third Int. atomic energy agency symposium on the physics and chemistry offission, Rochester, New Y ork, 1973 (International Atomic Energy Agency, Vienna, 1974) Vol. II, p. 273 2) J. R. Nix and A. J. Sierk, Phys. Scripta 10A (1974) 94 3) J. R. Nix and A. J. Sierk, Phys. Rev. Cl5 (1977) 2072 4) A. J. Sierk and J. R. Nix, Phys. Rev. Cl6 (1977) 1048 5) J. R. Nix, Nucl. Phys. A130 (1969); Lawrence Berkeley Laboratory Report No. UCRL-I7958 (1968), unpublished 6) K. T. R. Davies, A. J. Sierk and J. R. Nix, Phys. Rev. Cl3 (1976) 2385 7) S. E. Koonin, R. L. Hatch and J. Randrup, Nucl. Phys. A283 (1977) 87 8) S. E. Koonin and J. Randrup, Nucl. Phys. A289 (1977) 475 9) J. Blocki, Y. Boneh, J. R. Nix, J. Randrup, M. Robe], A. J. Sierk and W. J. Swiatecki, Ann. ofPhys. 113 (1978) 330 10) A. J. Sierk, S. E. Koonin and J. R. Nix, Phys. Rev. Cl7 (1978) 646 11) K. T. R. Davies, R. A. Managan, J. R. Nix and A. J. Sierk, Phys. Rev. Cl6 (1977) 1890 12) S. E. Koonin, K. T. R. Davies, V. Maruhn-Rezwani, H. Feldmeier, S. Krieger and J. W. Negele, Phys. Rev. Cl5 (1977) 1359 13) G. A. Pik-Pichak, JETP (Sov. Phys.) 16 (1963) 1201 14) H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1932) 6th ed., sect. 110, p. 147 15) Ibid., sect. 12, p. 12 16) Ibid., sect. 10, p. 8 17) Ibid., sect. 329, pp. 5799581 18) W. D. Myers and W. J. Swiatecki, Ark. Fys. 36 (1967) 343 19) G. Leander, Nucl. Phys. A219 (1974) 245 20) M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, New York, 1965) 5th printing, pp. 597-599 21) H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1932) 6th ed., sect. 320, p. 568

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R. L. Hatch and A. J. Sierk 1 Nuclear hydrodynamics

22) S. Cohen, F. Plasil and W. J. Swiatecki, Ann. of Phys. 82 (1974) 557 23) L. P. Meissner, Lawrence Berkeley Laboratory Computer Center Program No. D2 BJY ZAM (1965) unpublished 24) J. A. Zonneveld, Automatic numerical integration, Mathematical Center Tract No. 8 (Mathematrsch Centrum, Amsterdam, 1964) p. 23 25) R. Beringer and W. J. Knox, Phys. Rev. 121 (1961) II95 26) B. C. Carlson and Pao Lu, Proc. of the Rutherford Jubilee Int. Conf., Manchester, 1961 (Heywood, London, 1961), p. 291 27) B. Carnahan, H. A. Luther and J. 0. Wilkes, Applied numerical methods (Wiley, New York, 1969) sect. 5.9, pp. 319-320 28) S. Cohen and W. J. Swiatecki, Ann. of Phys. 22 (1963) 406 29) J. R. NIX, Ann. of Phys. 41 (1967) 52 30) A. E. S. Green, Nuclear physics (McGraw-Hill, New York, 1955) pp. 185, 250 31) S. Chandresekhar, Proc. Lond. Math. Sot. 9 (1959) I41 32) H. H. Tang and C. Y. Wong, J. of Phys. A7 (1974) 1038