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Some aspects of macroscopic nuclear dynamics
N U C L E A R I N S T R U M E N T S AND M E T H O D S
146 ( 1 9 7 7 )
213-223 ; ©
N O R T H - H O L L A N D P U B L I S H I N G CO.
SOME ASPECTS OF MACROSCOPIC NUCLEAR DYNAMICS* t J. R A N D R U P *
The Niels Bohr Institute, University ol Copenhagen, DK-2100 Copenhagen 0, Denmark
1. Introduction In a macroscopic approach to nuclear physics the interest is focused on the average properties of the phenomena, the aim being the revelation of general trends which can provide the natural background for a subsequent systematic study of the detailed behaviour of individual cases. The starting point is the fact (or claim) that the nucleon number is relatively large: A ~ 1. This idealization permits the use of statistical methods and justifies the neglect of effects arising from the discreteness of the nucleons. For nuclei an additional simplification is present which may be exploited to facilitate the analyses, namely that the surface can be considered relatively thin: b ,~ R (nuclei are leptodermous). One illustration of the macroscopic-leptodermous approach is the nuclear droplet model ~) which is known to account well for the average trends in nuclear energies and sizes, including the shape dependence of the nuclear potential energy. This talk is divided into two main chapters. The first chapter is concerned with the conservative forces acting between macroscopic leptodermous systems; this discussion will be based on the concept of proximity energy. The second chapter concerns the dissipative forces acting during the dynamical development of such a system, particularly those arising from the motion of the individual nucleons inside the deforming leptodermous container. These dissipative forces turn out so large that the inertial forces are expected to be of relatively minor importance and hence no third chapter dealing with these forces is included here. 2. The nuclear proximity energy The concept of proximity energy can be introduced in general terms by discussing the potential * Invited talk. t Based in part on work done in collaboration with J. Blocki, W. J. Swiatecki and C. F. Tsang. * On leave of absence from Aarhus University, Aarhus, Denmark.
energy of a leptodermous system2). This energy can be separated into a bulk part originating from the uniform interior and a part associated with the thin surface layer. It is instructive to expand the surface-layer energy in an asymptotic series of which the first term is proportional to the total surface area, the second term to the integrated surface curvature, and so on. However, due to the asymptotic nature of such an expansion, it is not possible in this way to include contributions to the potential energy arising from violent contortions in the surface, for example such that that elements of the surface face each other across a narrow gap or crevice. (Such a situation is encountered in the initial stages of a nuclear collision.) This additional energy is denoted the proximity energy. For a wide class of geometries of interest it is possible to derive simple approximative expressions for the proximity energy. 2.1. GENERALTREATMENT In the following is discussed the proximity en, ergy associated with a gap or a crevice whose width D is a slowly varying function of position. The proximity energy may then be approximated by Vp ,~ j e(O) do'.
(1)
Here e(D) is the interaction energy per unit area for two similar surfaces which are fiat and parallel and have the separation D. The integral is to.be carried out over the gap between the two interacting surfaces. This approximation procedure is illustrated in fig. 1. The above formula expresses the basic approximation made in calculating the proximity energy. By virtue of this approximation the problem separates into the determination of the key function e(D), which may be carried out once and for all (for a given type of material), and the subsequent gap integration involving only the geometry of the problem considered. ili.
NUCLEAR REACTION
MECHANISMS
214
j, R A N D R U P
where the function q~ is the incomplete integral of the proximity function ~o:
do-
t - 7 q~(~') d ~ ' .
The coefficient /~ measures the mean radius of curvature of the gap function. For two coaxial paraboloids with circular cross sections it is given by
v, = fe
/
R -1 = c ~ ' + c ; ' ,
Fig. 1. Illustration of the approximation procedure employed for the calculation of the proximity energy.
It is convenient to introduce as unit of length the surface width, b, and as unit of energy twice the specific surface energy, 27. For nuclei, b~lfm and 7 ~ , l M e V / f m 2. The key function e(D) may then be reduced to dimensionless form by writing
e(~b) = 2y ~o(0.
(2)
Thus ~ is the surface separation in units of b and q~ is the interaction energy per unit area in units of 2?'. The function ~0(0 shall henceforth be referred to as the universal proximity function. The qualitative behaviour of ~0 is easily inferred. For large separations it tends to zero rapidly because of the short range of the cohesive forces. As the two fiat surfaces are brought closer together, a mutual attraction develops and q~ becomes increasingly negative. At contact, which means that the two effective surface locations just coincide, the two matter distributions add up to approximate uniformity. This point shall be taken as zero point for the separation variable ~. At this point the two original surfaces have completely disappeared and consequently ~ ( 0 ) ~ - 1. It should be added that possible small deviations from uniformity have only a small effect on this value since the energy is stationary with respect to density variations around the bulk value. The pushing of the systems further into each other, still without permitting any density rearrangement will result in a strong repulsion and a correspondingly rapid increase of ~0. 2.2. PARABOLIC GAP
For many purposes it suffices to approximate the gap function D by a paraboloid. In this case the proximity energy may be written
Vp = 47rTbR~,
(4)
(3)
(5)
where Ca and C2 denote the radii of curvature of the two paraboloids. From the expression above it follows immediately by differentiation with respect to the separation that the force acting between the two curved surfaces is Fp =
4xT/~go.
(6)
As was discussed above, the universal function ~o has a minimum value approximately equal to minus one when the two surfaces are at contact. From this fact it follows as a general result that the maximum attraction between (any) two curved surfaces occurs when they are at contact and is given approximately by F,.~x ~
- 47t~R.
(7)
For example, one has for two spheres of radius R (approximated by paraboloids) made of mica (R = 5 cm) : Fmax = - 9300 dyn (= - 9 . 6 g weight); water (R = 5 mm): Fma x = - 70 dyn (= - 72 mg weigh0; nuclei (R = 5 fro) : F,,a, = - 3 1 MeV/fm (= - 500 kg weight). Cohesive forces between smooth curved surfaces of mica, rubber and gelatin have in fact been found experimentally to have this order of magnitude3,4), 2.3. THE NUCLEAR PROXIMITY FUNCTION A calculation of the nuclear proximity function has been carried out within the framework of the Thomas-Fermi approximation2). The density distributions of two semiinfinite systems of nuclear matter were superimposed without any rearrangements of the individual density distributions, which each correspond to the equilibrium state of one isolated surface. The kinetic energy is calculated by use of the Thomas-Fermi approximation and the interaction energy in the combined system is subsequently calculated on the basis of an effec-
MACROSCOPIC
NUCLEAR
tive two-nucleon interaction of the form ~--1"12/a
= - c
(1- P
dbb,
(8)
where P~2 is the relative nucleon momentum. This interaction was introduced into nuclear physics by Seyler and Blanchard 5) and it has proved very useful for the discussion of the gross nuclear energetics and sizes. For the parameter values a, b and C were chosen those determined by Myers and Swiatecki in their study of average nuclear propertiesb. The calculated proximity function ~0 is shown in fig. 2 and its incomplete integral q~ in fig. 3. The proximity function is seen to exhibit the qualitative features discussed earlier. The fact that the minimum value is not quite minus one is due to the surface scewness of the Thomas-Fermi density distribution. :,.. 0.6
=
I
0.4 The function .E 0.2
.g_
215
DYNAMICS
For the practical applications it is useful to have available an analytic representation of the potential. It turns out 2) that a good approximation to the integrated proximity function ~ (which is the one entering in the final expression) in the interesting region is given by / - ½ ( ~ - 2.54)z- 1.0852 ( ~ - 2.54) 3, ~ < 1.2511, ~(~) ~ [-3.437
~>1.2511.
e -{/O'75
(9) The approach described in the preceding makes it very simple to calculate approximately the nuclear potential between any two nuclei. An indication of the accuracy can be obtained by comparing with the results obtained with the energy densitfy formalism6). The present model, based on an effective interaction, and the energy density formalism, in which the energy is written as a functional of the local density, are expected to produce rather similar results. In figs. 4 and 5 are shown the nuclear potential for 4°Ar+121Sb and 84Kr+2°gBi as obtained by the energy density formalism together with the proximity results. The
o.o
= 2 -©.2
I
I
4020
~ -O.q
I
~
[
I
4OAr
1
I
I
I
+ 121Sb
i~ -o.~ -o.8
,ff=o i -I
-I._20
L
~
I
i
/
2
..,
N -20 4 - 40
{,the separation s in units of b
Fig. 2. The universal nuclear proximity function ~0(~) calculated on the basis of an effective interaction in the Thomas-Fer-
-60
'
I
'
~T---I \Proximity formula + T. F. I 5
mi approximation. ~= 2.0 C~ 'S o~ "=E 1.0 c to c .90.0
d et
I
'
I
'
I
I
The
I
function {o
/
L 6
i 7
i 8
L i 9 Io R(frn)
J It
I L2
L 13
Fig. 4. The nuclear potential energy for 4°Ar and 121Sb. The
full curve is the results obtained with the energy density forrealism 6) while the dots are given by the proximity formula. 40
I
~•~1 O I
I 8I 4 K r I +209Bi
10 • I
I
20
~ -i.o
o
> -20 ?: -2.o
•
-40 -60
*". -3.0- 4
-2'
_11 , the separgtion s in units of b
',
Fig. 3. The integrated proximity function ~b(¢) entering in the expression for the proximity energy associated with a parabolic gap.
Proximity formula +T.F.
-80
t 7
I 8
I 9
I I0
I II
1 i 12 15 R (fro)
i 14
I 15
16
Fig. 5. Similar to fig. 4, but for 84Kr and 2°9Bi.
Ill.
NUCLEAR
REACTION
MECHANISMS
216
J, R A N D R U P
overall good agreement verifies that expectation and, furthermore, indicates that the proximity formula is a reasonably good approximation to the exact six-dimensional interaction integral. [This is substantiated by a separate study with a semi-realistic analytically solvable modeiT).] Another way of comparing is to divide the various energy density potentials by the appropriate geometrical factor 4 7rybR. As required by the proximity formula, this leads to approximately the same function which is quite similar to the function ~ displayed in fig. 32). 2.4. EXPERIMENTAL DATA The experimental information on the proximity potential gathered so far is rather sparse and indirect. The best and most unambiguous information derives from elastic scattering data. In a recent analysisS), values of the nuclear potential were extracted from a number of elastic scattering experiments. The projectiles were B, C, N, O, Ne and S while the targets ranged from B to Pb. The analysis yields the value U of the nuclear potential at the distance of closest approach R for the classical orbit leading to the experimental rainbow angle. I
I
I
2.5. POSSIBLE EXTENSIONS
The nucleus-nucleus potential discussed above pertains to the idealized situation where all degrees of freedom except the relative motion are
1 • 2.,
.02
These data are displayed as dots in fig. 6. The abscissa is the surface separation calculated as s=R-C~-C~, where C j = R ~ - b 2 / R i with R j = l . 1 5 A ~ f m and b = l f m , file ordinate is the extracted potential U divided by the geometrical proximity factor 4 rcybR, where/~- I = C,- 1+ C2 l and the surface energy is taken as y = 1 - 2 [ ( N - Z ) / A ] 2 MeV/fm 2. The full-drawn curve is the theoretical proximity function ~ calculated as described earlier. On the whole it represents the average trend of the experimental points quite well. Large fluctuations around this smooth average are expected just as the nuclear binding energies exhibit fluctuations around the liquid-drop values. In addition, the underlying experiments as well as the subsequent data analysis are associated with some uncertainty which is difficult to ascertain. Finally it should be noted that the exponential approximation (9), which is indicated by the open squares, represents the exact function well over the range included.
°°°l
F, ~
.04
.O6 c o
"5 c'
.08 ~1
"
°a
.10
"~
8
.12
a/
~ = (R-C1-C2) /b
.14
.16
I
I
2.5
3.0
1
3.5 Surface separation
I
I
4.0
4.s
Fig. 6. Elastic scattering data. The values of the nuclear potential derived from experiment8) have been divided by the appropriate geometrical factor 4nTbR in order to obtain the "experimental" value of the universal function ~. The surface separation is obtained from the corresponding experimental value of the center separation R. Each dot represents one such set of elastic scattering data. The full curve is the calculated function q~ and the open squares indicate some values of the exponential approximation to 4'.
MACROSCOPIC
NUCLEAR
217
DYNAMICS
kept frozen. In a nuclear collision the system is quickly taken out of this very limited subspace. From the above general outline it is evident that it would be rather straightforward to include the additional degrees of freedom associated with the nuclear shape (this just amounts to modifying the geometrical factor) and even with the surface structure, e.g. the surface d~f.fuseness(which would require a corresponding recalculation of the universal proximity function). Another restrictive assumption made above was that the kinetic energies would readjust completely to abide with the exclusion principle. This corresponds to slow relative motion. For higher relative velocities, as may be encountered in the initial stages of a nuclear collision, a significant velocity dependence of the potential is expected, parly because of the gradual relaxation of the Pauli exclusion and partly because of the explicit velocity dependence of the effective nucleon-nucleon interaction. It turns out that these effects can be included in the proximity potential in a relatively simple manner but this aspect shall not be pursued here. Rather, I would like to indicate briefly how the frozen approach can be used as a simple tool for systematizing nuclear collisions.
of freedom associated with the nuclear shape but also the dissipative forces due to the irreversible coupling to the internal degrees of freedom. Not withstanding the necessity of introducing such complexity into a realistic treatment, the idealized treatment may serve as a useful guide as to what to expect in the initial stages of a collision process, when it is decided which general type of process will develop. The simple cubic approximation to the proximity function brings the discussion of nuclear collisions within the realm of analytical treatment. As a starting point for the analytical treatment, a characteristic length and a characteristic energy may be introducedg). The characteristic length may conveniently be taken as that separation where the proximity attraction sets in: the touching separation given by rt=C,+C2+2.54fm. This is the classical reaction separation. The characteristic energy is then taken as the interaction energy at that point, Et=e2Z, Z2/r,. From these two quantities may be formed two corresponding dimensionless parameters which characterize the particular case in question. The first is o = b/rt, the inverse of the touching separation in units of the nuclear surface diffuseness. The other is
2.6. NUCLEARCOLLISIONSIN THE FROZENIDEALIZATION fn the following is treated only the idealized case when all degrees of freedom except the relative motion are frozen. A realistic treatment would have to include not only additional degrees
X = e2 Z~Z22/4nTR.
0.15
I
'
I
'
I
r t
(10)
J
This is the ratio between the (repulsive) Coulomb force at touch and the maximum (attractive) nuclear force. For many systems of interest the par,
I
S,,S ,
S+O
0.10
Kr.
Ne ÷ Ni
S
Ni + Ni
Sn*~)
so..
•
Kr*Kr 0o
Mo+Kr •
Hf÷ArTh,A ~
E P 0.05
0.00
~ 0.0
1 0.f
~
I 0.2
L PerQmeter
I 0.3
~
I 0.4
, 0.5
X
Fig. 7. The position of a selection of target-projectile combinations in an X-o diagram. Ill.
NUCLEAR
REACTION
MECHANISMS
218
j RANDRUP
ameter o is approximately one tenth while X increases from 0.1 for two oxygen nuclei to above 1 for the heaviest combinations. In fig. 7 are displayed a number of target-projectile combinations in an X-t) diagram. It should be noted that there is a strong correlation between the two variables. The combined nuclear and Coulomb potential may now be written on a dimensionless form as V(r)/E t = I+VS--V
-~--V
Center separation R ( f m ) 6
7
8
9
10
11
12
Vp*Vc
].3
13
14
58Ni , 58Ni
t
"5 1.2
_c c o) ~ 0.9
+
"6 +v
( 1 - ~ ) 5+3v+ 2
....
(11)
where 6 = (rL--r)/b. As an illustration in fig. 8 the potential for two nickel nuclei is shown. In this figure is also indicated the effective potential corresponding to an angular momentum of L = 50. The cross section for two (frozen) nuclei approaching to a minimum separation rmm less than a certain specified separation r is given by 'r(rm,n < r) = ~r~ [1 -- V ( ~ ) / E ] ,
(12)
where E is the relative energy. In fig. 9 is plotted this cross section for two nickel nuclei, in units of a o = rcr~. (the classical reaction cross section at large energy). As can be seen from the X-~ diagram (fig. 7) the corresponding plot for sulphur and tin would be almost identical. The values of the characteristic quantities X, u, E, and a0 for 0.9
I
r
N a~ 0.8 0.7
~ -3
-2
;
0
I
2
3
4
5
6
Surface separation s Ifm)
Fig. 8. The potential energy for two frozen 58Ni nuclei as obtained by adding the proximity potential and the (point-charge) Coulomb potential. Also shown is the effective radial potential for an angular m o m e n t u m of 50 h.
some sample projectile-target combinations are given in table I. The correlation between X and u pointed out above implies that essentially only the one parameter X enters the discussion. This scaling feature is analogous to macroscopic fission theory, where properly scaled barriers, half-lives, fragment energies, etc. are primarily determined by the fissility parameter x. The analytical treatment can be carried further
I
r Srnin
r
4.0
0.8
SeNi ÷ 58Ni ¢
/
0.7
E, : 102.6 MeV
"5 0.6
O-o
3800 mb
==
X
0.35
.E
/
/
v OO9
0.5
-
J /
/
3.0
/
/
2 .54
/
212
(g 0.~ l/3
o
0.3
u
-23
0.2
~-2.5 0.1 0.0
I
0
0.5
1
1.5
Projectite energy in units of
2
2.5
Et
Fig. 9. Dimensionless scattering diagram for 58Ni on 58Ni, in the frozen approximation. The lines indicate the cross section for approaching closer than a given value of the surface separation rrnin. The nuclear attraction sets in at approximately s = 2.54 fro.
MACROSCOPIC
NUCLEAR
TABLE | Characteristic quantities for some projectile-target combinations. For a selection of target-projectile combinations are shown the values of the dimensionless parameters X and v as well as the values of the touching energy E t and the corresponding cross section a 0. X
v
Z60 + r60
0.098
0.131
12.05
1836
{~S + t60
0.141
0.118
21.76
2254
~S
+ ~S
E t (MeV)
% (rob)
0.201
0.108
39.66
2714
2010Ne+ 5828N~' l18q~ 500- + 160
0.208
0.105
42.14
2876
0.236
0.095
54.57
3501
36"" 84k'r + 3216 S 208m, 160 82,u + 118qn50 °'` + 3216 S
0.287
0.093
77.12
3634
0.295 0.339
0.085 0.088
80.33 101.23
4345 4069
L'sgr'ri + 28N158. 28
0.352
0.091
102.69
3797
180~4f72,,+ 4018Ar 84Kr + 36Kr 84
0.405
0.080
148.61
4954
0.411
0.082
152.77
4688
2~Th+
0.447
0.076
177.49
5427
0.451
0.080
174.83
4872
40Ar 18' • 96Mo+ ~4Kr
to give formulas for the depth of the potential pocket and the position and energy of the corresponding barrier, including the effect of the angular momentumS).
3. The nuclear one-body dissipation mechanism For nuclear collisions at energies above the Coulomb barrier an outstanding feature is the presence of dissipative processes. Most strikingly this is observed in the strongly damped or deep inelastic collisions where energies of several MeV per nucleon in the initial relative motion are converted into internal excitation. This large energy conversion takes place over just a few fm of relative separation implying that the braking forces acting amount to hundreds of MeV/fm. The experimental observation of these dissipation dominated processes constitutes a challenge to nuclear theory and demands an effort aimed at a better understanding of the underlying mechanisms, Generally speaking, the energy of the relative motion can be shared between all other degrees of freedom available in the total system. In the following, the interest is focused on the conversion into incoherent internal excitations. This conversion represents an irreversible flow of ordered collective energy into disordered microscopic heat
219
DYNAMICS
and has thus the character of true dissipation. Initial energy may also be lost to the various collective modes. Because of the coherent nature of such excitations this part of the energy loss is not truly dissipative and it may, in the final analysis, turn out to be necessary to include the collective modes explicitly. It is instructive to dinstinguish between two basic mechanisms for the dynamical production of the microscopic excitations. One consists of the inelastic collisions of two nucleons off each other and may be referred to as a direct mechanism involving the two-body interactions directly. By this mechanism the nucleons are scattered out of their initial orbitals into new unoccupied orbitals. This is the mechanism giving rise to ordinary viscosity, (The term direct used here to characterize this basic dissipation mechanism should not be confused with direct nucleus-nucleus reactions.) A contrasting mechanism is the irreversible excitation of the individual nucleons due to the time variation of the average nuclear field. In this case the dissipation is mediated by thes single-particle one-body potential and may thus be termed as indirect. An approximate estimate of the relative probability that a certain amount of excitation is produced by the two-body or the one-body mechanism is given by m) Wz(E)/WI(E ) ~ 10-3E (MeV). (13) Hence, for energies not too high above the Coulomb barrier the mean-field excitation mechanism is expected to play the dominant role. In the following only this mechanism will be discussed. A number of approaches have been taken towards the description of the mean-field dissipation mechanism. In simplest terms it may be stated that the basic mechanism by which the irreversible excitations are produced is the collision of the individual nucleons with the moving potential surface of the nucleus. This was first suggested by Beck and Gross LI) who derived a classical equation of motion with friction on the basis of the manybody Schfi3dinger equation. A related, but more general approach is represented by the adaption by Hofman and Siemens of linear response theory for heavy-ion collisionsn). The linear response theory treats the dynamical problem by first-order timedependent perturbation theory, relative to complete thermal equilibrium at a temperature which is determined simultaneously. This theory has recently been employed for the calculation of the response function for distant nuclear collisions~3). Ill. NUCLEAR
REACTION
MECHANISMS
220
j. R A N D R U P
It is important to point out that these approaches only cover the situations which are characterized by the slow time variation of the average potential. Such situations are encountered in nuclear fission and in distant nuclear collisions. For closer collisions a window or neck is suddently formed and the associated subsequent exchange of particles gives rise to a drastic increase in the energy loss and a corresponding increase in the complexity of a microscopic treatment. The situation can be discussed in relatively simple terms in a macroscopic framework of individual-particle motion in the nuclear mean field and as was pointed out by Swiatecki~4), the velocity mismatch between the exchanged nucleons leads to a friction force acting on the relative motion of the two nuclei. Detailed studies still remain to be carried out to determine whether this mechanism can account for the gross features as observed experimentally. 3.1. SECOND-ORDER KINETIC THEORY General insight into the mean-field excitation mechanism can be gained from a macroscopic treatment. In the most simple picture one may simply consider a classical gas of independent particles inside a deforming container. Such a model is justified by the facts that nuclei are leptodermous (the surface thickness is small as compared with the overall dimensions of the uniform interior) and that the nucleon mean free path is relatively long (as compared with the surface difusseness). The general formula governing the rate of energy change can be obtained as follows. Consider a small element da of the container surface (sufficiently small that it may be considered planar). The velocity of this part of the container surface in the normal direction is denoted by rE The moving wall gives rise to forces in the normal direction only so that the problem separates into the unaffected motion parallel to the surface and the motion in the normal direction. All that enters is then the local density of particles having a velocity component v, in the normal direction, p(v,). Such particles collide with the surface at a rate given by o(v~)=p(v~)(vn-n)0(~--Vn). The truncation function expresses that only particles with v~>h collide with the surface. In the frame of the moving surface, the particles reflect elastically by reversing their normal velocity. It follows that the observed energy change per collision is
Ae(vn)=-2mr~(v,-r~). The total rate of energy change is found by integrating over the local normal velocity and subsequently over the whole surface,
=-2mr ,daf,
dvnp(vn)(v,-h) 2.
(14)
This is the general formula valid for any velocity distribution. In the special case when the velocity distribution is uniform and isotropic, within a sphere of radius vF, and constant along the surface, the above expression becomes
= - mpo ff1@
hda 1 -
1+~-
With the assumption that the surface speed is small as compared with the typical particle speed an expansion may be performed in the corresponding ratio of speeds. To second order the genral expression (14) above may then be written
F, ,~ - m _f h da p(v~) + 2m .f
h2
da p(v,).
(16)
Here (v,) denotes the (local) average of the particle speed in the direction towards the wall and (@ denotes the (local) average of the square of that speed (which is proportional to the pressure on the wall). In this general expression the first term, being linear in the surface velocity, represents the reversible energy change associated with an adiabatic change of the surface shape. It may be integrated up to give the potential energy of the system. It is simply the ideal gas law stating that the kinetic energy (equivalent to the temperature) is proportional to the pressure times the volume. Note that for volume-conserving shape changes this term vanishes (if the pressure can be considered constant along the surface). The second term is always positive. It represents the irreversible excitation caused by the dynamical change of the container shape. It governs the rate of dissipative flow from the collective shape degrees of freedom into internal heat. Its form as a surface integral should be contrasted with the volume character of ordinary viscosity produced by the two-body interactions between the constituents (which is the dominant dissipation mechansire in ordinary fluids). Thus it gives rise to a whole new type of dynamics which may be the important one for the discussion of nuclear systems in the intermediate-energy region above the low-
MACROSCOPIC
NUCLEAR
TABLE 2
The amount of damping for various multipole modes due to the one-body dissipation mechanism for a number of nuclei. The damping coefficient is measured in units of critical damping for an idealized drop. From ref. 15. n=2
n=4
12o~ 50~,*
3.01 5.24
1.94 2.88
1.38 1.94
0.99 1.37
23811 92~
9.46
3.58
2.27
1.58
~0°Ne
n=8
n=16
temperature superfluid region and below the highenergy two-body dominated region. 3.2.
T H E NUCLEAR RESPONSE TO MULTIPOLE DISTORTIONS
The characteristic features of the nuclear dissipation implied by the above mechanism appear quite different from those of ordinary viscosity. A simple schematlc illustration of the basic differences is provided by the dissipation produced in nuclei by multipole-type distortions. (These distortions are not supposed to represent ordinary lowtemperature vibrations but serve the purpose of clarifying the response to shape changes characteristic ,of a heated nucleus.) Such a study has been carried out by Blocki iS) who calculated the amount of damping induced by the shape motion. The result is conveniently expressed relative to the critical damping calculated on the basis of the nuclear liquid-drop model. The result is listed in table 2. The most striking feature is that generally the motion is overdamped. This is at variance with the two-body viscosity coefficients usually employed. Moreover, the damping is largest for gentle distortions (low multipole order), also contrary to usual viscosity. Furthermore, the overdamping increases with the size of the system. 3.3. FISSION WITH ONE-BODY VISCOSITY Taking the preceding illustration as a general indication of the features associated with the onebody viscosity one can make some qualitative inferences for the nuclear fission process, Firstly, one would expect the dynamical motion to procede in a creeping manner due to the overdamping. Hence one would not expect intertiai forces to be important. Secondly, the direction taken in distortion space is determined by the relative sizes of
221
DYNAMICS
the ratios between the conservative forces and the dissipative forces for the various distortion types. This would lead to the expectation that the neck formation will develop faster than the overall elongation as the former type of distortion is considerably less damped than the latter type. As a consequence of this dynamical development the scission shapes encountered should be appreciably more compact than what would result from calculations ~6) with the ordinary two-body viscosity. With the two-body viscosity the damping is smallest for the gentle distortions and the fission will proceed by stretching rather than necking leading to rather elongated scission shapes. An additional general feature expected for the one-body dissipation is its weak temperature dependence, also contrary to two-body viscosity. This should be seen in light of the experimental finding that the fission fragment kinetic energies appear independent of the excitation energy up to above 100 MeV. The above qualitative expectations are borne out by actual calculationtT). It is found that the scis300
~
..... 250
I ~--'
i
I
. . . .
1
. . . .
NONVISCOUS / ~' INFINITE IWO-BOOY VISCOSITY / ONE- BODY DISSIPATION, / FERMI-GAS VALUE
0 200 Z
bJ
~-
Z, / ,'
~5o
°/.
]
z
<,T- I00
,
//
.-"'"
...........
u') Z < cK 1--
50
0
, J ~
I
500
,
~
,
~
I
J
iooo Z 2/A I/3
t
,
,
I
~500
L
J
,
,
000
Fig. 10. T h e translational kinetic energies of fission f r a g m e n t s , from ref. 17. T h e upper curve results from a dynamical calculation without d a m p i n g while t h e lower curve corresponds to an infinite viscosity o f t h e two-body type. T h e middle curve is obtained with the one-body dissipation formula as given in the text. T h e experimental results are also included.
111. N U C L E A R
REACTION
MECHANISMS
222
J. RANDRUP
sion shapes are in fact relatively compact. The overdamped nature of the motion is confirmed by the fact that the dynamical path is rather insensitive to the actual value of the damping coefficient. The results obtained for the fission fragment kinetic energies are shown in fig. 10. The excellent reproduction of the average trend of the experimental results, without any parameter adjustment, is very encouraging and urges the pursuit of the implications of the one-body dissipation mechanism for nuclear collision dynamics.
4. Concluding remarks This talk has dealt partly with the potential energy between two distinct nuclear systems. In such case very simple analytical formulas can be derived. The assumption of having two distinct systems has most validity in the initial stages of a nuclear collision. In a realistic dynamical treatment the inclusion of other macroscopic degrees of freedom is mandatory. Particularly the neck formation is expected to be of major importance. The experinaental measurement of the nucleus-nucleus potential is obscured by the presence of large dissipative effects. At present, the only relatively clear-cut information about the potential derives from elastic scattering data and hence is confined to the tail region. In this region the analytical proximity formula appears a good average of the values extracted from experiment. A better understanding of the dissipation phenomena is required before the interpretation of the deeper-probing experimental results is reliable. For energies not too far above the Coulomb barrier the dissipation is expected to be dominated by the time variation of the nuclei~r mean field. In the macroscopic approach simple formulas can be derived for the rate of energy flow out of the collective motion into incoherent internal excitation. It turns out that this dissipation is very large. In fact the predicted dissipative forces are so large that inertial forces may be neglected. This introduces a substantial simplification into the discussion. Experimental evidence from nuclear fission lends support to the assumption that the nuclear dynamics is dominated by the one-body dissipation mechanism but no final conclusions should yet be drawn. The dynamical implications of the one-body dissipation for nuclear collisions remain to be studied carefully. It seems likely that the analytical treatment can be carried far and one may
hope that a very simple description of the gross dynamical features will emerge. The complexity of the nuclear collision phenomena makes it difficult to test the theoretical models and a general warning should be made about not drawing premature conclusions. Usually many different phenomena interplay in one single collision and the fact that a certain model may be made to fit the available experimental data should not be taken as a guarantee that it is necessarily correct. A lot of theoretical work still remains before a clear understanding is reached of the many aspects involved. For the experimentalists much work needs to be carried out in order to provide sufficient information to make clear discriminations possible between the various theoretical interpretations. For this job the tandem accelerator plays an important role as a flexible and accurate tool. 1 would like to thank the organizers of the conference for the invitation to participate. Furthermore, I am grateful to my colleagues at the Niels Bohr Institute, particularly J. P. Bondorf and P. J. Siemens, for helpful discussions during the preparation of the manuscript. Most of the basic ideas discussed here are due to W. J. Swiatecki and 1 am very grateful to him for many inspiring discussions during the work. Finally, I wish to thank J. Blocki, P. R. Christensen and J. R. Nix for profitable discussions and for making their results available prior to publication.
References l) W. D. Myers and W. J. Swiatecki, Ann. Phys. 55 (1969) 395. 2) j. Blocki, J. Randrup, W. J. Swiatecki and C. F. Tsang, Ann. Phys. 105 (1977) 427. 3) j. N. lsraelachvili and D. Tabor, Proc. Roy. Soc. London A331 (1972) 19. 4) K. L. Johnson, K. Kendall and A. O. Roberts, Proc. Roy. Soc. London A324 (1971) 301. 5) R. G. Seyler and C. H. Blanchard, Phys. Rev, 124 (1961) 227; 131 (1963)355. 6) C. Ng6, B. Tamain, M. Beiner, R. J. Lombard, D. Mas and H. H. Deubler, Nucl. Phys. A252 (1975) 237. 7) j. Randrup, Lawrence Berkeley Laboratory Report LBL-4017 (1975). 8) p. R. Christensen, private communication (1976). 9) W. J. Swiatecki, Int. School-Seminar on Reactions o/heaEv ions with nuclei and synthesis o[new elements, Dubna (1975). 10) D. H. E. Gross, Nucl, Phys. A240 (1975) 472, il) R. Beck and D. H. E. Gross, Phys. Lett. 47B (1973) 143. 12) H. Hofman and P. J. Siemens, Nucl. Phys. A257 (1976) 165.
MACROSCOPIC NUCLEAR DYNAMICS 13) p. j. Johanson, A. S. Jensen, P. J. Siemens and H. Hofmann, to be published. 14) W. J. Swiatecki, J. Physique C5 (1972) 45. 15) j. Blocki, private communication (1975). 16) j. R. Nix and A. Sierk, Int. Workshop II1 on Gross proper-
223
ties oi nuclei and nuclear excitations, Hirschegg, Austria (1975). 17) A. J. Sierk and J. R. Nix, LAP-151, Syrup. on Macroscopic .leatures of heavy-ion colfisions. Argonne National Laboratory, Argonne, Illinois (1976).
II1. N U C L E A R R E A C T I O N MECHANISMS
146 ( 1 9 7 7 )
213-223 ; ©
N O R T H - H O L L A N D P U B L I S H I N G CO.
SOME ASPECTS OF MACROSCOPIC NUCLEAR DYNAMICS* t J. R A N D R U P *
The Niels Bohr Institute, University ol Copenhagen, DK-2100 Copenhagen 0, Denmark
1. Introduction In a macroscopic approach to nuclear physics the interest is focused on the average properties of the phenomena, the aim being the revelation of general trends which can provide the natural background for a subsequent systematic study of the detailed behaviour of individual cases. The starting point is the fact (or claim) that the nucleon number is relatively large: A ~ 1. This idealization permits the use of statistical methods and justifies the neglect of effects arising from the discreteness of the nucleons. For nuclei an additional simplification is present which may be exploited to facilitate the analyses, namely that the surface can be considered relatively thin: b ,~ R (nuclei are leptodermous). One illustration of the macroscopic-leptodermous approach is the nuclear droplet model ~) which is known to account well for the average trends in nuclear energies and sizes, including the shape dependence of the nuclear potential energy. This talk is divided into two main chapters. The first chapter is concerned with the conservative forces acting between macroscopic leptodermous systems; this discussion will be based on the concept of proximity energy. The second chapter concerns the dissipative forces acting during the dynamical development of such a system, particularly those arising from the motion of the individual nucleons inside the deforming leptodermous container. These dissipative forces turn out so large that the inertial forces are expected to be of relatively minor importance and hence no third chapter dealing with these forces is included here. 2. The nuclear proximity energy The concept of proximity energy can be introduced in general terms by discussing the potential * Invited talk. t Based in part on work done in collaboration with J. Blocki, W. J. Swiatecki and C. F. Tsang. * On leave of absence from Aarhus University, Aarhus, Denmark.
energy of a leptodermous system2). This energy can be separated into a bulk part originating from the uniform interior and a part associated with the thin surface layer. It is instructive to expand the surface-layer energy in an asymptotic series of which the first term is proportional to the total surface area, the second term to the integrated surface curvature, and so on. However, due to the asymptotic nature of such an expansion, it is not possible in this way to include contributions to the potential energy arising from violent contortions in the surface, for example such that that elements of the surface face each other across a narrow gap or crevice. (Such a situation is encountered in the initial stages of a nuclear collision.) This additional energy is denoted the proximity energy. For a wide class of geometries of interest it is possible to derive simple approximative expressions for the proximity energy. 2.1. GENERALTREATMENT In the following is discussed the proximity en, ergy associated with a gap or a crevice whose width D is a slowly varying function of position. The proximity energy may then be approximated by Vp ,~ j e(O) do'.
(1)
Here e(D) is the interaction energy per unit area for two similar surfaces which are fiat and parallel and have the separation D. The integral is to.be carried out over the gap between the two interacting surfaces. This approximation procedure is illustrated in fig. 1. The above formula expresses the basic approximation made in calculating the proximity energy. By virtue of this approximation the problem separates into the determination of the key function e(D), which may be carried out once and for all (for a given type of material), and the subsequent gap integration involving only the geometry of the problem considered. ili.
NUCLEAR REACTION
MECHANISMS
214
j, R A N D R U P
where the function q~ is the incomplete integral of the proximity function ~o:
do-
t - 7 q~(~') d ~ ' .
The coefficient /~ measures the mean radius of curvature of the gap function. For two coaxial paraboloids with circular cross sections it is given by
v, = fe
/
R -1 = c ~ ' + c ; ' ,
Fig. 1. Illustration of the approximation procedure employed for the calculation of the proximity energy.
It is convenient to introduce as unit of length the surface width, b, and as unit of energy twice the specific surface energy, 27. For nuclei, b~lfm and 7 ~ , l M e V / f m 2. The key function e(D) may then be reduced to dimensionless form by writing
e(~b) = 2y ~o(0.
(2)
Thus ~ is the surface separation in units of b and q~ is the interaction energy per unit area in units of 2?'. The function ~0(0 shall henceforth be referred to as the universal proximity function. The qualitative behaviour of ~0 is easily inferred. For large separations it tends to zero rapidly because of the short range of the cohesive forces. As the two fiat surfaces are brought closer together, a mutual attraction develops and q~ becomes increasingly negative. At contact, which means that the two effective surface locations just coincide, the two matter distributions add up to approximate uniformity. This point shall be taken as zero point for the separation variable ~. At this point the two original surfaces have completely disappeared and consequently ~ ( 0 ) ~ - 1. It should be added that possible small deviations from uniformity have only a small effect on this value since the energy is stationary with respect to density variations around the bulk value. The pushing of the systems further into each other, still without permitting any density rearrangement will result in a strong repulsion and a correspondingly rapid increase of ~0. 2.2. PARABOLIC GAP
For many purposes it suffices to approximate the gap function D by a paraboloid. In this case the proximity energy may be written
Vp = 47rTbR~,
(4)
(3)
(5)
where Ca and C2 denote the radii of curvature of the two paraboloids. From the expression above it follows immediately by differentiation with respect to the separation that the force acting between the two curved surfaces is Fp =
4xT/~go.
(6)
As was discussed above, the universal function ~o has a minimum value approximately equal to minus one when the two surfaces are at contact. From this fact it follows as a general result that the maximum attraction between (any) two curved surfaces occurs when they are at contact and is given approximately by F,.~x ~
- 47t~R.
(7)
For example, one has for two spheres of radius R (approximated by paraboloids) made of mica (R = 5 cm) : Fmax = - 9300 dyn (= - 9 . 6 g weight); water (R = 5 mm): Fma x = - 70 dyn (= - 72 mg weigh0; nuclei (R = 5 fro) : F,,a, = - 3 1 MeV/fm (= - 500 kg weight). Cohesive forces between smooth curved surfaces of mica, rubber and gelatin have in fact been found experimentally to have this order of magnitude3,4), 2.3. THE NUCLEAR PROXIMITY FUNCTION A calculation of the nuclear proximity function has been carried out within the framework of the Thomas-Fermi approximation2). The density distributions of two semiinfinite systems of nuclear matter were superimposed without any rearrangements of the individual density distributions, which each correspond to the equilibrium state of one isolated surface. The kinetic energy is calculated by use of the Thomas-Fermi approximation and the interaction energy in the combined system is subsequently calculated on the basis of an effec-
MACROSCOPIC
NUCLEAR
tive two-nucleon interaction of the form ~--1"12/a
= - c
(1- P
dbb,
(8)
where P~2 is the relative nucleon momentum. This interaction was introduced into nuclear physics by Seyler and Blanchard 5) and it has proved very useful for the discussion of the gross nuclear energetics and sizes. For the parameter values a, b and C were chosen those determined by Myers and Swiatecki in their study of average nuclear propertiesb. The calculated proximity function ~0 is shown in fig. 2 and its incomplete integral q~ in fig. 3. The proximity function is seen to exhibit the qualitative features discussed earlier. The fact that the minimum value is not quite minus one is due to the surface scewness of the Thomas-Fermi density distribution. :,.. 0.6
=
I
0.4 The function .E 0.2
.g_
215
DYNAMICS
For the practical applications it is useful to have available an analytic representation of the potential. It turns out 2) that a good approximation to the integrated proximity function ~ (which is the one entering in the final expression) in the interesting region is given by / - ½ ( ~ - 2.54)z- 1.0852 ( ~ - 2.54) 3, ~ < 1.2511, ~(~) ~ [-3.437
~>1.2511.
e -{/O'75
(9) The approach described in the preceding makes it very simple to calculate approximately the nuclear potential between any two nuclei. An indication of the accuracy can be obtained by comparing with the results obtained with the energy densitfy formalism6). The present model, based on an effective interaction, and the energy density formalism, in which the energy is written as a functional of the local density, are expected to produce rather similar results. In figs. 4 and 5 are shown the nuclear potential for 4°Ar+121Sb and 84Kr+2°gBi as obtained by the energy density formalism together with the proximity results. The
o.o
= 2 -©.2
I
I
4020
~ -O.q
I
~
[
I
4OAr
1
I
I
I
+ 121Sb
i~ -o.~ -o.8
,ff=o i -I
-I._20
L
~
I
i
/
2
..,
N -20 4 - 40
{,the separation s in units of b
Fig. 2. The universal nuclear proximity function ~0(~) calculated on the basis of an effective interaction in the Thomas-Fer-
-60
'
I
'
~T---I \Proximity formula + T. F. I 5
mi approximation. ~= 2.0 C~ 'S o~ "=E 1.0 c to c .90.0
d et
I
'
I
'
I
I
The
I
function {o
/
L 6
i 7
i 8
L i 9 Io R(frn)
J It
I L2
L 13
Fig. 4. The nuclear potential energy for 4°Ar and 121Sb. The
full curve is the results obtained with the energy density forrealism 6) while the dots are given by the proximity formula. 40
I
~•~1 O I
I 8I 4 K r I +209Bi
10 • I
I
20
~ -i.o
o
> -20 ?: -2.o
•
-40 -60
*". -3.0- 4
-2'
_11 , the separgtion s in units of b
',
Fig. 3. The integrated proximity function ~b(¢) entering in the expression for the proximity energy associated with a parabolic gap.
Proximity formula +T.F.
-80
t 7
I 8
I 9
I I0
I II
1 i 12 15 R (fro)
i 14
I 15
16
Fig. 5. Similar to fig. 4, but for 84Kr and 2°9Bi.
Ill.
NUCLEAR
REACTION
MECHANISMS
216
J, R A N D R U P
overall good agreement verifies that expectation and, furthermore, indicates that the proximity formula is a reasonably good approximation to the exact six-dimensional interaction integral. [This is substantiated by a separate study with a semi-realistic analytically solvable modeiT).] Another way of comparing is to divide the various energy density potentials by the appropriate geometrical factor 4 7rybR. As required by the proximity formula, this leads to approximately the same function which is quite similar to the function ~ displayed in fig. 32). 2.4. EXPERIMENTAL DATA The experimental information on the proximity potential gathered so far is rather sparse and indirect. The best and most unambiguous information derives from elastic scattering data. In a recent analysisS), values of the nuclear potential were extracted from a number of elastic scattering experiments. The projectiles were B, C, N, O, Ne and S while the targets ranged from B to Pb. The analysis yields the value U of the nuclear potential at the distance of closest approach R for the classical orbit leading to the experimental rainbow angle. I
I
I
2.5. POSSIBLE EXTENSIONS
The nucleus-nucleus potential discussed above pertains to the idealized situation where all degrees of freedom except the relative motion are
1 • 2.,
.02
These data are displayed as dots in fig. 6. The abscissa is the surface separation calculated as s=R-C~-C~, where C j = R ~ - b 2 / R i with R j = l . 1 5 A ~ f m and b = l f m , file ordinate is the extracted potential U divided by the geometrical proximity factor 4 rcybR, where/~- I = C,- 1+ C2 l and the surface energy is taken as y = 1 - 2 [ ( N - Z ) / A ] 2 MeV/fm 2. The full-drawn curve is the theoretical proximity function ~ calculated as described earlier. On the whole it represents the average trend of the experimental points quite well. Large fluctuations around this smooth average are expected just as the nuclear binding energies exhibit fluctuations around the liquid-drop values. In addition, the underlying experiments as well as the subsequent data analysis are associated with some uncertainty which is difficult to ascertain. Finally it should be noted that the exponential approximation (9), which is indicated by the open squares, represents the exact function well over the range included.
°°°l
F, ~
.04
.O6 c o
"5 c'
.08 ~1
"
°a
.10
"~
8
.12
a/
~ = (R-C1-C2) /b
.14
.16
I
I
2.5
3.0
1
3.5 Surface separation
I
I
4.0
4.s
Fig. 6. Elastic scattering data. The values of the nuclear potential derived from experiment8) have been divided by the appropriate geometrical factor 4nTbR in order to obtain the "experimental" value of the universal function ~. The surface separation is obtained from the corresponding experimental value of the center separation R. Each dot represents one such set of elastic scattering data. The full curve is the calculated function q~ and the open squares indicate some values of the exponential approximation to 4'.
MACROSCOPIC
NUCLEAR
217
DYNAMICS
kept frozen. In a nuclear collision the system is quickly taken out of this very limited subspace. From the above general outline it is evident that it would be rather straightforward to include the additional degrees of freedom associated with the nuclear shape (this just amounts to modifying the geometrical factor) and even with the surface structure, e.g. the surface d~f.fuseness(which would require a corresponding recalculation of the universal proximity function). Another restrictive assumption made above was that the kinetic energies would readjust completely to abide with the exclusion principle. This corresponds to slow relative motion. For higher relative velocities, as may be encountered in the initial stages of a nuclear collision, a significant velocity dependence of the potential is expected, parly because of the gradual relaxation of the Pauli exclusion and partly because of the explicit velocity dependence of the effective nucleon-nucleon interaction. It turns out that these effects can be included in the proximity potential in a relatively simple manner but this aspect shall not be pursued here. Rather, I would like to indicate briefly how the frozen approach can be used as a simple tool for systematizing nuclear collisions.
of freedom associated with the nuclear shape but also the dissipative forces due to the irreversible coupling to the internal degrees of freedom. Not withstanding the necessity of introducing such complexity into a realistic treatment, the idealized treatment may serve as a useful guide as to what to expect in the initial stages of a collision process, when it is decided which general type of process will develop. The simple cubic approximation to the proximity function brings the discussion of nuclear collisions within the realm of analytical treatment. As a starting point for the analytical treatment, a characteristic length and a characteristic energy may be introducedg). The characteristic length may conveniently be taken as that separation where the proximity attraction sets in: the touching separation given by rt=C,+C2+2.54fm. This is the classical reaction separation. The characteristic energy is then taken as the interaction energy at that point, Et=e2Z, Z2/r,. From these two quantities may be formed two corresponding dimensionless parameters which characterize the particular case in question. The first is o = b/rt, the inverse of the touching separation in units of the nuclear surface diffuseness. The other is
2.6. NUCLEARCOLLISIONSIN THE FROZENIDEALIZATION fn the following is treated only the idealized case when all degrees of freedom except the relative motion are frozen. A realistic treatment would have to include not only additional degrees
X = e2 Z~Z22/4nTR.
0.15
I
'
I
'
I
r t
(10)
J
This is the ratio between the (repulsive) Coulomb force at touch and the maximum (attractive) nuclear force. For many systems of interest the par,
I
S,,S ,
S+O
0.10
Kr.
Ne ÷ Ni
S
Ni + Ni
Sn*~)
so..
•
Kr*Kr 0o
Mo+Kr •
Hf÷ArTh,A ~
E P 0.05
0.00
~ 0.0
1 0.f
~
I 0.2
L PerQmeter
I 0.3
~
I 0.4
, 0.5
X
Fig. 7. The position of a selection of target-projectile combinations in an X-o diagram. Ill.
NUCLEAR
REACTION
MECHANISMS
218
j RANDRUP
ameter o is approximately one tenth while X increases from 0.1 for two oxygen nuclei to above 1 for the heaviest combinations. In fig. 7 are displayed a number of target-projectile combinations in an X-t) diagram. It should be noted that there is a strong correlation between the two variables. The combined nuclear and Coulomb potential may now be written on a dimensionless form as V(r)/E t = I+VS--V
-~--V
Center separation R ( f m ) 6
7
8
9
10
11
12
Vp*Vc
].3
13
14
58Ni , 58Ni
t
"5 1.2
_c c o) ~ 0.9
+
"6 +v
( 1 - ~ ) 5+3v+ 2
....
(11)
where 6 = (rL--r)/b. As an illustration in fig. 8 the potential for two nickel nuclei is shown. In this figure is also indicated the effective potential corresponding to an angular momentum of L = 50. The cross section for two (frozen) nuclei approaching to a minimum separation rmm less than a certain specified separation r is given by 'r(rm,n < r) = ~r~ [1 -- V ( ~ ) / E ] ,
(12)
where E is the relative energy. In fig. 9 is plotted this cross section for two nickel nuclei, in units of a o = rcr~. (the classical reaction cross section at large energy). As can be seen from the X-~ diagram (fig. 7) the corresponding plot for sulphur and tin would be almost identical. The values of the characteristic quantities X, u, E, and a0 for 0.9
I
r
N a~ 0.8 0.7
~ -3
-2
;
0
I
2
3
4
5
6
Surface separation s Ifm)
Fig. 8. The potential energy for two frozen 58Ni nuclei as obtained by adding the proximity potential and the (point-charge) Coulomb potential. Also shown is the effective radial potential for an angular m o m e n t u m of 50 h.
some sample projectile-target combinations are given in table I. The correlation between X and u pointed out above implies that essentially only the one parameter X enters the discussion. This scaling feature is analogous to macroscopic fission theory, where properly scaled barriers, half-lives, fragment energies, etc. are primarily determined by the fissility parameter x. The analytical treatment can be carried further
I
r Srnin
r
4.0
0.8
SeNi ÷ 58Ni ¢
/
0.7
E, : 102.6 MeV
"5 0.6
O-o
3800 mb
==
X
0.35
.E
/
/
v OO9
0.5
-
J /
/
3.0
/
/
2 .54
/
212
(g 0.~ l/3
o
0.3
u
-23
0.2
~-2.5 0.1 0.0
I
0
0.5
1
1.5
Projectite energy in units of
2
2.5
Et
Fig. 9. Dimensionless scattering diagram for 58Ni on 58Ni, in the frozen approximation. The lines indicate the cross section for approaching closer than a given value of the surface separation rrnin. The nuclear attraction sets in at approximately s = 2.54 fro.
MACROSCOPIC
NUCLEAR
TABLE | Characteristic quantities for some projectile-target combinations. For a selection of target-projectile combinations are shown the values of the dimensionless parameters X and v as well as the values of the touching energy E t and the corresponding cross section a 0. X
v
Z60 + r60
0.098
0.131
12.05
1836
{~S + t60
0.141
0.118
21.76
2254
~S
+ ~S
E t (MeV)
% (rob)
0.201
0.108
39.66
2714
2010Ne+ 5828N~' l18q~ 500- + 160
0.208
0.105
42.14
2876
0.236
0.095
54.57
3501
36"" 84k'r + 3216 S 208m, 160 82,u + 118qn50 °'` + 3216 S
0.287
0.093
77.12
3634
0.295 0.339
0.085 0.088
80.33 101.23
4345 4069
L'sgr'ri + 28N158. 28
0.352
0.091
102.69
3797
180~4f72,,+ 4018Ar 84Kr + 36Kr 84
0.405
0.080
148.61
4954
0.411
0.082
152.77
4688
2~Th+
0.447
0.076
177.49
5427
0.451
0.080
174.83
4872
40Ar 18' • 96Mo+ ~4Kr
to give formulas for the depth of the potential pocket and the position and energy of the corresponding barrier, including the effect of the angular momentumS).
3. The nuclear one-body dissipation mechanism For nuclear collisions at energies above the Coulomb barrier an outstanding feature is the presence of dissipative processes. Most strikingly this is observed in the strongly damped or deep inelastic collisions where energies of several MeV per nucleon in the initial relative motion are converted into internal excitation. This large energy conversion takes place over just a few fm of relative separation implying that the braking forces acting amount to hundreds of MeV/fm. The experimental observation of these dissipation dominated processes constitutes a challenge to nuclear theory and demands an effort aimed at a better understanding of the underlying mechanisms, Generally speaking, the energy of the relative motion can be shared between all other degrees of freedom available in the total system. In the following, the interest is focused on the conversion into incoherent internal excitations. This conversion represents an irreversible flow of ordered collective energy into disordered microscopic heat
219
DYNAMICS
and has thus the character of true dissipation. Initial energy may also be lost to the various collective modes. Because of the coherent nature of such excitations this part of the energy loss is not truly dissipative and it may, in the final analysis, turn out to be necessary to include the collective modes explicitly. It is instructive to dinstinguish between two basic mechanisms for the dynamical production of the microscopic excitations. One consists of the inelastic collisions of two nucleons off each other and may be referred to as a direct mechanism involving the two-body interactions directly. By this mechanism the nucleons are scattered out of their initial orbitals into new unoccupied orbitals. This is the mechanism giving rise to ordinary viscosity, (The term direct used here to characterize this basic dissipation mechanism should not be confused with direct nucleus-nucleus reactions.) A contrasting mechanism is the irreversible excitation of the individual nucleons due to the time variation of the average nuclear field. In this case the dissipation is mediated by thes single-particle one-body potential and may thus be termed as indirect. An approximate estimate of the relative probability that a certain amount of excitation is produced by the two-body or the one-body mechanism is given by m) Wz(E)/WI(E ) ~ 10-3E (MeV). (13) Hence, for energies not too high above the Coulomb barrier the mean-field excitation mechanism is expected to play the dominant role. In the following only this mechanism will be discussed. A number of approaches have been taken towards the description of the mean-field dissipation mechanism. In simplest terms it may be stated that the basic mechanism by which the irreversible excitations are produced is the collision of the individual nucleons with the moving potential surface of the nucleus. This was first suggested by Beck and Gross LI) who derived a classical equation of motion with friction on the basis of the manybody Schfi3dinger equation. A related, but more general approach is represented by the adaption by Hofman and Siemens of linear response theory for heavy-ion collisionsn). The linear response theory treats the dynamical problem by first-order timedependent perturbation theory, relative to complete thermal equilibrium at a temperature which is determined simultaneously. This theory has recently been employed for the calculation of the response function for distant nuclear collisions~3). Ill. NUCLEAR
REACTION
MECHANISMS
220
j. R A N D R U P
It is important to point out that these approaches only cover the situations which are characterized by the slow time variation of the average potential. Such situations are encountered in nuclear fission and in distant nuclear collisions. For closer collisions a window or neck is suddently formed and the associated subsequent exchange of particles gives rise to a drastic increase in the energy loss and a corresponding increase in the complexity of a microscopic treatment. The situation can be discussed in relatively simple terms in a macroscopic framework of individual-particle motion in the nuclear mean field and as was pointed out by Swiatecki~4), the velocity mismatch between the exchanged nucleons leads to a friction force acting on the relative motion of the two nuclei. Detailed studies still remain to be carried out to determine whether this mechanism can account for the gross features as observed experimentally. 3.1. SECOND-ORDER KINETIC THEORY General insight into the mean-field excitation mechanism can be gained from a macroscopic treatment. In the most simple picture one may simply consider a classical gas of independent particles inside a deforming container. Such a model is justified by the facts that nuclei are leptodermous (the surface thickness is small as compared with the overall dimensions of the uniform interior) and that the nucleon mean free path is relatively long (as compared with the surface difusseness). The general formula governing the rate of energy change can be obtained as follows. Consider a small element da of the container surface (sufficiently small that it may be considered planar). The velocity of this part of the container surface in the normal direction is denoted by rE The moving wall gives rise to forces in the normal direction only so that the problem separates into the unaffected motion parallel to the surface and the motion in the normal direction. All that enters is then the local density of particles having a velocity component v, in the normal direction, p(v,). Such particles collide with the surface at a rate given by o(v~)=p(v~)(vn-n)0(~--Vn). The truncation function expresses that only particles with v~>h collide with the surface. In the frame of the moving surface, the particles reflect elastically by reversing their normal velocity. It follows that the observed energy change per collision is
Ae(vn)=-2mr~(v,-r~). The total rate of energy change is found by integrating over the local normal velocity and subsequently over the whole surface,
=-2mr ,daf,
dvnp(vn)(v,-h) 2.
(14)
This is the general formula valid for any velocity distribution. In the special case when the velocity distribution is uniform and isotropic, within a sphere of radius vF, and constant along the surface, the above expression becomes
= - mpo ff1@
hda 1 -
1+~-
With the assumption that the surface speed is small as compared with the typical particle speed an expansion may be performed in the corresponding ratio of speeds. To second order the genral expression (14) above may then be written
F, ,~ - m _f h da p(v~) + 2m .f
h2
da p(v,).
(16)
Here (v,) denotes the (local) average of the particle speed in the direction towards the wall and (@ denotes the (local) average of the square of that speed (which is proportional to the pressure on the wall). In this general expression the first term, being linear in the surface velocity, represents the reversible energy change associated with an adiabatic change of the surface shape. It may be integrated up to give the potential energy of the system. It is simply the ideal gas law stating that the kinetic energy (equivalent to the temperature) is proportional to the pressure times the volume. Note that for volume-conserving shape changes this term vanishes (if the pressure can be considered constant along the surface). The second term is always positive. It represents the irreversible excitation caused by the dynamical change of the container shape. It governs the rate of dissipative flow from the collective shape degrees of freedom into internal heat. Its form as a surface integral should be contrasted with the volume character of ordinary viscosity produced by the two-body interactions between the constituents (which is the dominant dissipation mechansire in ordinary fluids). Thus it gives rise to a whole new type of dynamics which may be the important one for the discussion of nuclear systems in the intermediate-energy region above the low-
MACROSCOPIC
NUCLEAR
TABLE 2
The amount of damping for various multipole modes due to the one-body dissipation mechanism for a number of nuclei. The damping coefficient is measured in units of critical damping for an idealized drop. From ref. 15. n=2
n=4
12o~ 50~,*
3.01 5.24
1.94 2.88
1.38 1.94
0.99 1.37
23811 92~
9.46
3.58
2.27
1.58
~0°Ne
n=8
n=16
temperature superfluid region and below the highenergy two-body dominated region. 3.2.
T H E NUCLEAR RESPONSE TO MULTIPOLE DISTORTIONS
The characteristic features of the nuclear dissipation implied by the above mechanism appear quite different from those of ordinary viscosity. A simple schematlc illustration of the basic differences is provided by the dissipation produced in nuclei by multipole-type distortions. (These distortions are not supposed to represent ordinary lowtemperature vibrations but serve the purpose of clarifying the response to shape changes characteristic ,of a heated nucleus.) Such a study has been carried out by Blocki iS) who calculated the amount of damping induced by the shape motion. The result is conveniently expressed relative to the critical damping calculated on the basis of the nuclear liquid-drop model. The result is listed in table 2. The most striking feature is that generally the motion is overdamped. This is at variance with the two-body viscosity coefficients usually employed. Moreover, the damping is largest for gentle distortions (low multipole order), also contrary to usual viscosity. Furthermore, the overdamping increases with the size of the system. 3.3. FISSION WITH ONE-BODY VISCOSITY Taking the preceding illustration as a general indication of the features associated with the onebody viscosity one can make some qualitative inferences for the nuclear fission process, Firstly, one would expect the dynamical motion to procede in a creeping manner due to the overdamping. Hence one would not expect intertiai forces to be important. Secondly, the direction taken in distortion space is determined by the relative sizes of
221
DYNAMICS
the ratios between the conservative forces and the dissipative forces for the various distortion types. This would lead to the expectation that the neck formation will develop faster than the overall elongation as the former type of distortion is considerably less damped than the latter type. As a consequence of this dynamical development the scission shapes encountered should be appreciably more compact than what would result from calculations ~6) with the ordinary two-body viscosity. With the two-body viscosity the damping is smallest for the gentle distortions and the fission will proceed by stretching rather than necking leading to rather elongated scission shapes. An additional general feature expected for the one-body dissipation is its weak temperature dependence, also contrary to two-body viscosity. This should be seen in light of the experimental finding that the fission fragment kinetic energies appear independent of the excitation energy up to above 100 MeV. The above qualitative expectations are borne out by actual calculationtT). It is found that the scis300
~
..... 250
I ~--'
i
I
. . . .
1
. . . .
NONVISCOUS / ~' INFINITE IWO-BOOY VISCOSITY / ONE- BODY DISSIPATION, / FERMI-GAS VALUE
0 200 Z
bJ
~-
Z, / ,'
~5o
°/.
]
z
<,T- I00
,
//
.-"'"
...........
u') Z < cK 1--
50
0
, J ~
I
500
,
~
,
~
I
J
iooo Z 2/A I/3
t
,
,
I
~500
L
J
,
,
000
Fig. 10. T h e translational kinetic energies of fission f r a g m e n t s , from ref. 17. T h e upper curve results from a dynamical calculation without d a m p i n g while t h e lower curve corresponds to an infinite viscosity o f t h e two-body type. T h e middle curve is obtained with the one-body dissipation formula as given in the text. T h e experimental results are also included.
111. N U C L E A R
REACTION
MECHANISMS
222
J. RANDRUP
sion shapes are in fact relatively compact. The overdamped nature of the motion is confirmed by the fact that the dynamical path is rather insensitive to the actual value of the damping coefficient. The results obtained for the fission fragment kinetic energies are shown in fig. 10. The excellent reproduction of the average trend of the experimental results, without any parameter adjustment, is very encouraging and urges the pursuit of the implications of the one-body dissipation mechanism for nuclear collision dynamics.
4. Concluding remarks This talk has dealt partly with the potential energy between two distinct nuclear systems. In such case very simple analytical formulas can be derived. The assumption of having two distinct systems has most validity in the initial stages of a nuclear collision. In a realistic dynamical treatment the inclusion of other macroscopic degrees of freedom is mandatory. Particularly the neck formation is expected to be of major importance. The experinaental measurement of the nucleus-nucleus potential is obscured by the presence of large dissipative effects. At present, the only relatively clear-cut information about the potential derives from elastic scattering data and hence is confined to the tail region. In this region the analytical proximity formula appears a good average of the values extracted from experiment. A better understanding of the dissipation phenomena is required before the interpretation of the deeper-probing experimental results is reliable. For energies not too far above the Coulomb barrier the dissipation is expected to be dominated by the time variation of the nuclei~r mean field. In the macroscopic approach simple formulas can be derived for the rate of energy flow out of the collective motion into incoherent internal excitation. It turns out that this dissipation is very large. In fact the predicted dissipative forces are so large that inertial forces may be neglected. This introduces a substantial simplification into the discussion. Experimental evidence from nuclear fission lends support to the assumption that the nuclear dynamics is dominated by the one-body dissipation mechanism but no final conclusions should yet be drawn. The dynamical implications of the one-body dissipation for nuclear collisions remain to be studied carefully. It seems likely that the analytical treatment can be carried far and one may
hope that a very simple description of the gross dynamical features will emerge. The complexity of the nuclear collision phenomena makes it difficult to test the theoretical models and a general warning should be made about not drawing premature conclusions. Usually many different phenomena interplay in one single collision and the fact that a certain model may be made to fit the available experimental data should not be taken as a guarantee that it is necessarily correct. A lot of theoretical work still remains before a clear understanding is reached of the many aspects involved. For the experimentalists much work needs to be carried out in order to provide sufficient information to make clear discriminations possible between the various theoretical interpretations. For this job the tandem accelerator plays an important role as a flexible and accurate tool. 1 would like to thank the organizers of the conference for the invitation to participate. Furthermore, I am grateful to my colleagues at the Niels Bohr Institute, particularly J. P. Bondorf and P. J. Siemens, for helpful discussions during the preparation of the manuscript. Most of the basic ideas discussed here are due to W. J. Swiatecki and 1 am very grateful to him for many inspiring discussions during the work. Finally, I wish to thank J. Blocki, P. R. Christensen and J. R. Nix for profitable discussions and for making their results available prior to publication.
References l) W. D. Myers and W. J. Swiatecki, Ann. Phys. 55 (1969) 395. 2) j. Blocki, J. Randrup, W. J. Swiatecki and C. F. Tsang, Ann. Phys. 105 (1977) 427. 3) j. N. lsraelachvili and D. Tabor, Proc. Roy. Soc. London A331 (1972) 19. 4) K. L. Johnson, K. Kendall and A. O. Roberts, Proc. Roy. Soc. London A324 (1971) 301. 5) R. G. Seyler and C. H. Blanchard, Phys. Rev, 124 (1961) 227; 131 (1963)355. 6) C. Ng6, B. Tamain, M. Beiner, R. J. Lombard, D. Mas and H. H. Deubler, Nucl. Phys. A252 (1975) 237. 7) j. Randrup, Lawrence Berkeley Laboratory Report LBL-4017 (1975). 8) p. R. Christensen, private communication (1976). 9) W. J. Swiatecki, Int. School-Seminar on Reactions o/heaEv ions with nuclei and synthesis o[new elements, Dubna (1975). 10) D. H. E. Gross, Nucl, Phys. A240 (1975) 472, il) R. Beck and D. H. E. Gross, Phys. Lett. 47B (1973) 143. 12) H. Hofman and P. J. Siemens, Nucl. Phys. A257 (1976) 165.
MACROSCOPIC NUCLEAR DYNAMICS 13) p. j. Johanson, A. S. Jensen, P. J. Siemens and H. Hofmann, to be published. 14) W. J. Swiatecki, J. Physique C5 (1972) 45. 15) j. Blocki, private communication (1975). 16) j. R. Nix and A. Sierk, Int. Workshop II1 on Gross proper-
223
ties oi nuclei and nuclear excitations, Hirschegg, Austria (1975). 17) A. J. Sierk and J. R. Nix, LAP-151, Syrup. on Macroscopic .leatures of heavy-ion colfisions. Argonne National Laboratory, Argonne, Illinois (1976).
II1. N U C L E A R R E A C T I O N MECHANISMS