A model for calculating the angular momentum distribution of fission fragments

Nuclear Physics Al36 (1969) 465480;

Not to be reproduced by photoprint

@ North-Holland Publishing Co., Amsterdam

or microfilm without written permission from the publisher

A MODEL FOR CALCULATING THE ANGULAR MOMENTUM DISTRIBUTION OF FISSION FRAGMENTS

Institut fiir Xheoretische Physik der Universitiit Heidelberg, Germany and & J. MANG

Received Xl June 1969 Abstract: A scheme is presented for quantitative calculations of the distribution of rotational angular momenta in primary fission fragments. The wave function of the bending mode excitation at the scission point is expanded in angular momentum components. As an example, the distribution of rotationaf angular momentum is estimated for the fragment of mass 108 from the thermal neutron fission of z39Pu. The experimental fiterature on the subject is reviewed,

1. Introduction The population of rotational states in alpha decay of deformed even nuclei essentially measures the angular form of the alpha-~robah~l~ty function near the nuclear surface “)* The amplitudes for alpha groups are the Legendre expansion coefficients of the angular wave packet, which, if it is highly concentrated in angle, implies much of high rotational components. If at the s&ion stage in the fission process3 we can estimate the wave function in the orientation of the deformed fragments, we can similarly expand in rotator i& functions. We must then consider what further modifications in the distribution might come subsequently by non-central interactions after scission, and we have the final distribution. From heavy-ion elastic scattering, we know the optical potential parameters between heavy nuclei at least in the grazing region. We assume that at s&&on we can approximate the energy of the system as that of deformed fragments interacting by an optical-model potential. 2. Fission dynamics In low-energy fission, the compound nucleus has practicaliy no excitation energy at the saddle point. Its energy is almost pure deformation energy. For a discussion of the fission process in the region between the saddle and the scission point, the 465

466

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et al.

RASMUSSEN

molecular model of fission “) is most suited. In this model, the nucleus is pictured as consisting of two interacting deformed fragments separated by the centre-to-centre distance 6. The potential energy v(a) of the ground state is assumed to depend on D as given in fig. 1. The scission barrier is indicated “) in molecular-model calculations for 240Pu. At some distance between the saddle and the scission point, the absolute value of the derivative av,,/&r of the attractive nuclear interaction between the fragments becomes larger than the absolute value of the derivative av,/&r of the repulsive Coulomb interaction because the nuclear interaction is strong and of short range. The fragments look like cigars (positive or prolate deformations) inside of the scission barrier. They will after scission tend by some sort of damped shape oscillations toward oblate equilibrium shapes. Of course, the assumption of the scission energy

I

:a,

I saddle point

distance

CT

c

between fragment centers-of- mass

interaction

Fig. 1. Schematic

diagram of the potential energy of the compound nucleus distance u between the fragment centres of mass.

as a function of the

barrier is h~othetical. But in any case, the potential energy should at least be flattened in the region. With the optical potential employed by Auerbach and Porter “) for heavy-ion scattering, a valley does develop inside the barrier. Fig. 2 is from their work; it lends plausibility to the picture of a scission valley as in the schematic diagram of fig. 1. Certainly the relatively low values of the average kinetic energy in fission imply strong damping of the ordered fission motion past the saddle point with the energy U (see fig. 1) largely going into random internal excitation “). Since the collective states have much larger interaction matrix elements than non-collective states and since these non-collective excitations are hindered by an energy gap “) of at least 2 MeV we expect that a statistical equilibrium at scission should only be established between collective degrees of freedom in contrast to the statistical model introduced by Fong ‘), who assumes a statistical equilibrium between all states. The essential point is that we

FISSION

467

FRAGMENTS

expect very little kinetic energy in the ordered fission mode up to the point at which the overlap of the fragments becomes negligible. The details of the fission mechanism near the scission point depend on the question whether the height of the scission barrier is small compared with U (case a) or not (case b). In the former case (a), it is reasonable to assume that the thermal equilibrium of collective states occurring most probably in the scission valley is not seriously altered during the rapid passage of the scission barrier. In case (b) of a high scission barrier, the system will tend to minimize the energy in the modes orthogonal to the fission mode when passing the barrier.

140.0

-

120.0

-

100.0

-

60.0

-

W=16.4 MeV R= l.26(Al”‘+A2n)fm 0=0.49fm .\ ‘\

POTENTIAL (MeV)

-40.01&& 0 Fig. 2. Potentials

3.0

6.0

, , , ,q 9.0

I20

15.0

of the “best fit” of the 197Au(‘60, 160) 19’Au data from ref. 4).

Therefore, also the energy in the modes of angular vibration of the deformed fragments [the “bending” and “wriggling” modes of Nix “)I is minimized, and only the ground states of these modes should be appreciably populated at scission. In this paper, we shall study explicitly only the latter case (b) with a high scission barrier. As is discussed at the end of sect. 5, the formulae derived for case (b) approximately hold for case (a) too with only slight changes of parameters. It is difficult, especially when one considers the diffuse tails of the nuclear matter outside of the nuclei, to define the point of scission without being arbitrary. To avoid as much of this arbitrariness as possible, we assume that the maximum of the scission barrier at which the system minimizes bending energy occurs at a centre-to-

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centre separation distance a, given uniquely by the kinetic energy of the fission event in question, Z,Z,e’ E kin = ___ cc

.

Experiments on angular distribution of fission fragments ‘) have shown that the component of total angular momentum along the symmetry axis (K) selected at the saddle point, is approximately conserved during the fission process from saddle to scission. This was first pointed out by Bohr lo). The coupling of different collective degrees of freedom past the saddle point is therefore restricted to states with equal quantum numbers K.Note that the Coriolis coupling of states with different values of K is rather weak because of the large moment of inertia of the compound nucleus ll). 3. The model for angular momentum distributions For the sake of simplicity, we consider only fission events with one undeformed fragment A, since otherwise the number of relevant angular variables and consequent geometrical complications is excessive. We presume that of the fission events producing one fragment A near doubly magic 132Sn some will have that fragment almost spherical at scission. We write the electrostatic potential energy V, of such a system including quadrupole and octupole distortions of the deformed fragment B. We ignore higher multipoles in distortion. This is seen to be justified from numerical studies on equilibrium deformations of two interacting fragments “). The deformation parameters u2 and cl3 define the surface of a uniformly charged fragment out to a surface R&9) = R,[l + a2 Pz(cos 0) + cc3

P,(cos

O)].

Then V,(a, y) z ZAZBe2 [I+

i (!9202P2(cos

7)+ ; (9a,P3(cos

Y)] ’

(1)

a

where y is the angle between the symmetry axis of the deformed fragment and the vector of the centre-to-centre distance a of the fragments. As usual, ZA and Za are the atomic numbers of the fragments. Note that Z,Z,e’/a may be approximately equated to the fission kinetic energy Ekin . Henceforth this substitution will be made. We assume that the attractive nuclear optical potential V, is a function of the closest distance s of the surfaces of the two fragments, and that we need take only the first two terms of the expansion about the scission barrier (s = 0)

We must now work out the dependence of s on the angle y between the symmetry axis of the deformed fragment and the centre-to-centre vector. Only small values of y

FISSION

469

FRAGMENTS

will be encountered; hence we approximate the deformed surface at its tip by a tangent sphere as in fig. 3. By the law of cosines, we have S = (~7’+ u2 -2au cos y)+. For small angles, we can expand as follows: S = [a2+u2-2au(l-sin2 Sz(a-u)

y)*]*,

1.

l+J 2 (a_U),sin2y+ _E_-

m

fragment

fragment

B

(2)

...

[ A

S

/

\a

s

R4

’

._

u--e

Fig. 3. Molecular

By a straightforward tangent sphere is

model diagram of the compound

geometrical calculation,

nucleus.

the distance u to the centre of the

3Ra( 1+ a2 + a&Q + 201,)

ll=

1+4a2+7a,

(3)

’

We avoid the arbitrariness of the parameters of the nuclear optical potential V, by assuming that the attractive nuclear force and the repulsive electrostatic force just balance each other at the scission point. Thus, from eq. (1) we get

--=

zAz:e2 [I+ 5(%)‘a2+ y ($)3a3]

.

At fixed a, then, we get the potential energy terms dependent on y of the following form:

=

ibin -& -*E,,,

[*+ g (%)‘a2+

[g e)2a2+

7

‘7’(:)‘a31 sin2y ($)3a3]

Sin27

= -)K sin’ y, where we have taken just the leading sin2y terms in V,. If we want to be very careful,

410

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et

al.

we must make a distinction between the R, in the first or nuclear term and the R, in the electrostatic second term. The former should be about 1.4 A* fm according to the optical-model analyses of Auerbach and Porter 4), who show the barrier top (fig. 2) for lg7Au+160 at about 11.7fm (i.e. 11.7 = 1.4 (197*+16*). The latter RB should have the value 1.2A* fm for the equivalent uniform charge radius. We use the larger radius throughout. Hence, K = Ekin k

[I-

; (; -2)e)‘fX2-

; (3;

-5)(%)3L?s]

.

(4)

Analogous to the alpha-decay problem for deformed even nuclei, the inertial parameter B for the bending motion is a reduced moment-of-inertia as follows: B-’

= 4-‘+(M,02)-‘,

(5)

where 9 is the rotational moment of inertia for the deformed fragment and M, the reduced mass of the two fragments. Thus for this rotational mode [a special case of the wriggling mode or bending mode of Nix *)I, we have a Hamiltonian 2

H = $L2++Ksin2y,

where

The Schriidinger equation with this Hamiltonian is the well-studied equation occurring in oblate spheroidal coordinates 12). We make the substitution ?.J= cos y, assume a q-function of the form eimQand get the standard form

with c2=_.

BK h2

Numerous formulae and tables of these functions are given by Abramowitz and Stegun I’). We shall see later that we are concerned with large c-values (x 50) where the asymptotic formulae for eigenvalues are useful. The leading term of the asymptotic formulae shows an approach to a spectrum of equally spaced eigenvalues with energy spacing FI(K/B)~. Eq. (21.8.2) of ref. 12) gives the asymptotic expansion for the eigenvalues. With c = 50, we get eigenvalues E,,,( = A,,,,,+c2 measured with respect to the potential minimum at y = 0): so0 = 99, slr = 198, se2 = 295, s22 = 297 etc. The spectrum is close to that of a two-dimensional harmonic oscillator. Indeed, if

FISSION FRAGMENTS

471

we replace sin y by y in eq. (6), we get in good approximation the Hamiltonian of a two-dimensional harmonic oscillator. The solution of the Schriidinger equation gives the eigenvalues E,,,,, = hw,(2n + /ml + l),

(8)

with the quantum energy of the bending mode hw, = WW)+,

(9)

and the quantum numbers n = 0, 1,2, . . . and m = +O, + 1, . . . . The eigenfunctions are given by

(10) with N,, = i [rcn!(n + Iml)!]-*, y; = h(KB)-+.

(11)

It was shown for alpha decay of spin-zero systems that a complete expansion in proper angular momentum functions of rotational and orbital angular momentum reduces to an expansion in the angle with a reduced moment of inertia B appearing ‘). For these spin-zero systems, m has to be zero. In general for systems with spin, the bending mode wave function u,,, can be expanded into spherical harmonics Y,, (y, q) to give the angular momentum distribution of this wave function. To get the final distribution immediately after scission, these angular momenta must be coupled with the residual spin of the fragment. The expansion coefficients are determined by the following integral:

or after the integration over cp has been carried out (lmlnm)

=

(_

l)“yo

(2z+l)(z-m)!

a

n!(n+Iml)!(Z+m)!

For y. < 1, we can let the upper limit go to infinity. Approximating polynomial for m 2 0 by a Bessel function according to ref. 13), Iqcos y) =

-1 m(Z+m)! 5,(3(2~+l)Y), l+_t (l-m)!

( 1

the Legendre

472 we

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0.

RASMUSSEN

et d.

obtain (lmlnrn)

~

( P+lN+m)!

(_I>”

*

n!(n+Iml)!(Z--m)!

xyt+l exp(-t(l+3)‘y~)L~((l+3)‘y~). Thus the probability are given by

distributions

I(lmlnm)12x

of angular momentum in the eigenstates (n, m) (21+ 1)(1+ Iml)! n!(n+ Iml)!(l- Iml)!

x

(ygp +l exp(-(~+~)‘Y~)C~,“‘((~+~)~Y~)I~. (12)

For the ground-state wave function, this expression reduces to at 3 1(10100)12z (2Z+l)y% exp(-(Z+3)2yz).

(13)

Generally one has for spin-zero systems with m = 0 a; E I(ZOlnO)12 w -iP+

n.t

lhitexp(-(z+~)2y~)CL~((z+3)2~~112. (134

The quantity 4/r,, is plotted in fig. 4 as a function of (Z+$)y, for II = 0, 1 and 2. The expression we have derived here has the same form as the distribution function proposed by Warhanek and Vandenbosch 14) on statistical grounds. We determine the average spin value approximately from eq. (13) by replacing the sum over 1 by an integral and obtain Jn 1,” = co -3. (14)

0.8

0

1

2

3

4

Fig. 4. Probability distribution of angular momenta for the bending states (n, m) = (1, 0), (2,0), (3,0). Curves derived from exact diagonalization of the Hamiltonian eq. (6) in the basis of angular momentum eigenfunctions would appear to be identical in this diagram for y0 s 0.2.

We have confined ourselves to case (b) of the fission mechanism near the scission point as discussed in sect. 2. Since in this case the bending mode energy is minimized,

FISSION

473

FRAGMENTS

we have only to consider the ground state, (n, m) = (0, 0). Thus eq. (13) gives the spin distribution and eq. (14) the average spin value of the deformed fragment immediately after scission. Deviations which occur for case (a) of the fission mechanism are discussed at the end of sect. 5.

4. Vibrational and rotational motion after scission It is necessary now to consider whether this distribution of angular momenta just after scission will be strongly altered by subsequent Coulomb excitation. The question is more difficult to formulate than for alpha decay, since the shapes of the fission fragments will usually not remain constant during the period when the electric multipole interaction is significant. In the limiting case that the deformed fragment keeps its pear shape or even only the quadrupole deformation, the Coulomb excitation effect could be quite large. The effect might best be treated by left-multiplying the a, vector of eq. (13) by a square FrSman 15) matrix. Since in contrast to alpha decay, fission will be over the barrier after scission, the argument of the Froman matrix will be mainly imaginary and not real. Strutinsky 16) has published such calculations in 1959. Hoffman 17) obtained experimental evidence for fragment primary spins and also carried out extensive classical numerical calculations of the torque integrals on a spheroidal fission fragment. One trouble, of course, in comparing these angular momenta from Coulomb excitation with experiment was that the initial boundary conditions at scission were not known. What our paper purports to do is to give the means of calculating these boundary conditions quantally. For convenience, we give here a simple classical estimate of the Coulomb excitation effect, although the calculations of Hoffman provide a more careful estimate. We suppose that the deformed fragment B has a stable quadrupole moment Qs and that the average angle between its symmetry axis and the fission axis is ye. During the time the fragments are accelerated by the repulsive Coulomb force, the fragment B with the quadrupole moment QB feels a torque which depends on time. Integrating over the time and dividing by h, we get the classical change Al of angular momentum in units of h Al =

s

dt ‘%.!.?& sin ye cos y,, . Wt)

Provided y. << 1 and with a(t) = we

[o-z+ gyt]‘,

get Al = (+eZM,ZA)*

‘$ . c

(15)

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et

d.

Consider now the opposite limiting case of undamped shape oscillations with a period short compared with the time for the fragments to move apart by a significant fraction of the separation distance at scission. In this case, the torques altering rotational angular momentum will average to zero. Nix Is) has numerically solved the classical equations of motion for separation of undamped vibrating spheroids, and one could use his solutions to estimate rotational Coulomb excitation. However, we feel that the assumption of undamped shape vibrations is probably quite unrealistic. We will content ourselves here with another classical estimate of the quadrupole Coulomb excitation that might occur for a critically damped shape vibration going to a spherical equilibrium. From eq. (l), we calculate the classical torque aV,,,,/ay for angle y. and then multiply by the characteristic time w;’ for the quadrupole vibration to get the classical change of angular momentum (in units of ti) 2

ct2(R0J-1.

Finally, one or both nuclei might oscillate one half cycle into an oblate shape, the equi~brium. In this case, the torque during the time in prolate shape might nearly be cancelled by that during the time spent in oblate shape. In considering the above questions, it has occurred to us that shell effects similar to those responsible for the ns-ms spontaneous fission isomers in Pu and Am might be of importance. In particular, we would suggest a new approach to understanding fission asymmetry, the puzzling dip in average kinetic energy near symmetric fission and the unequal distributions of excitation energy between fragments as revealed by the average number of neutrons emitted from each fragment (for a review see Hyde I’), subsect. 11.6). Strutinsky’s figs. 2 and 3 of ref. 2o) show contour maps of the deviation of Nilsson level densities as a function of nuclear number 2 (or N) and deformation ? = 20 (&I,,-- ~~i~)~~~. The shaded areas characterize lowest level density and consequently shell stabilized regions. Of course, doubly magic 13’Sn is shell stabilized near spherical shape on both the proton and the neutron diagram and analyses of the mean number V of emitted neutrons, such as Terrell’s ““) (fig. 5), suggest that fragments near doubly magic have little deformation energy at scission. For the complementary fragment in 239Pu+n fission (i.e. 1‘*RUDE), the shell stabilization valley for 64 neutrons is around q = 9 for both figs. 2 and 3 of ref. ‘“). Strutinsky’s proton diagram of fig. 2 shows that an q = 9-10 deformation might also be most favourable for 44 protons. Note that as the neutron number approaches 73 (corresponding to exact symmetric fission), the valley of stabilization near 71= 10 terminates for both neutrons and protons. Thus, the decreased yield and decreased kinetic energy in fission as one moves toward symmetric division could be due to the loss of shell stab~ization of the light fragment at f = 9 deformation and of the heavy fragment around zero de-

FISSION

475

FRAGMENTS

formation. The best compromise for a near symmetric split may be for the heavy fragment to move out to the next valley of shell stabilization beginning around q = 7. The centre-to-centre distance at the scission barrier would thus become 6,

I FISSION

0'. 70

,..I

00

I

I

I

I

I

150

I60

170

NEUTRONS

l

,

90

100

II0 INITIAL

I20

:

I30

FRAGMENT

Fig. 5. Terrell’s results *I) on neutron

140 MASS

multiplicitiesin fission.

---AL

120

130 Mass

140 number,

150

160

AH-

Fig. 6. Mass-energy contour for Pu fission from Milton and Fraser n3).

abruptly larger on approach to symmetry, and the kinetic energy in fission would consequently drop as experimentally observed (see fig. 6).

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5. Numerical results

We choose the thermal neutron fission of 23gPu as an example for two main reasons. Since the ground state of 23gPu has spin and parity tf. one of the fission channels of the excited 240Pu nucleus will be O’, and our theory is presently specialized to this case. Furthermore, the complementary fragment to doubly magic 132Sn is lo8Ru, which is in a region where evidence indicates stable deformation ‘“). With a deformed product and as a consequence a ground state rotational band, there is a greater likelihood that the angular momentum distribution in the fission fragment can be experimentally measured either by angular correlation or by gamma intensities of the rotational cascade. Fig. 6 is a mass energy contour for Pu fission taken from Milton and Fraser 23). For the mass split we consider, one obtains a most probable kinetic energy of about 180 MeV. From this, we calculate a centre-to-centre distance ’ at the scission barrier Z,Z,e’ B, = ~

= 17.6fm.

E kin

The denominator should, of course, really be the monopole-monopole Coulomb energy. To the extent that the tail of the nuclear potential is present at the scission barrier, we have underestimated the denominator by a few MeV. However, if the light fragment actually has a stable prolate deformation, we have overestimated the denominator by x 4 MeV (the monopole-quadrupole interaction). We consider the above effects may roughly cancel. As stated earlier, optical-model work on heavy-ion scattering indicates we should use a radius constant of r. = 1.40 fm. At the scission barrier then, we have 0, = ro[A$+&(1+cc2+cr3)]. With AA = 132 and A, = 108, we get O!,+Ct3

=

0.57.

This value is consistent with recent molecular-model studies by Dickmann and Dietrich ‘“) on the fission of 236U . Neglecting the nuclear interaction between the fragments, the potential energies for two tangent spheroidal fragments are calculated. For lo4Mo, a value of 2.1 for the axis ratio of the sphercid at scission is obtained. We do not know the ratio of u2 to u3 and feel that any liquid-drop-model estimate would be unreliable. Fortunately, the final answers are not too sensitive to this ratio 7 Nix has recently carried out ljquid-drop-model calculations 24) for heavy nuclei out to scission shapes. We cannot directly use his results since he deals with the class of reflection-symmetric figures of revolution only. He has calculated 31) that the centre-to-centre distance at scission for “‘U is about 22.5 fm. Thus, he has = 20 MeV of translational kinetic energy in the fission mode as the nucleus passes his scission point. His model gives the neck as cones apex-to-apex at scission. Were his picture to be corrected by rounding the physically unreasonable sharp tips, the actual scission point would occur at a smaller centre-to-centre distance more closely in accord with our postulate that the translational kinetic energy in the fission mode is small at the scission barrier.

417

FISSIONFRAGMENTS

as is seen from the two cases we consider, namely (I) a2 = a3 and (II) a2 = 2a3. Table 1 gives the results. Note that ye is essentially the reciprocal of the r.m.s. spin value. It is interesting that the az values imply deformations in the 9 = 9 region of shell stabilization alluded to in the discussion at the end of sect. 4. Angular momentum Quantity

aa

u3 oc II K B nwb

Yo &” at scission Coulomb excitation corrections: static quadrupole Al damped vibration Al

TABLE1 distribution parameters

for iosRu

u2 = a3

Case II 012= 2a3

0.285 0.285 17.6 fm 5.55 fm 74.1 MeV . radb2 310.095 6’ MeV-’

0.38 0.19 17.6 fm 5.31 fm 68.4MeV=rad-2 3jO.095 fi2 MeV-’

1.53 MeV 0.144 rad

1.46 MeV 0.147 rad

eq. (9) eq. (11)

5.6

5.5

eq. (14)

5.6

5.7

eq. (15) assuming Qt,=4b

1.9

2.5

eq. (16) assuming &02 = I MeV

Case I

Notes

eq. (3) eq. (4) eq. (5) and taking&+ (ii*Ru, ref. “)) “)

= 95 keV

“) Note adden in proof: The matter of the energy of the 2+ state in ii”Ru is now controversial. Zicha et al. 32) assign an energy of 97.7 keV, in agreement with Watson 22). Knowing of these measurements, Wilhelmy 33) has carefully examined his data on short-lived fission products and data of colleagues on prompt fission-coincident y-rays. In this way he assigns a level at 240.7 keV with half-life 0.24 nsec as the 2+ state. WilheImy’s data also cast doubt on the ground rotational band assignments of Zicha et a!. for neigh~ring Ru nucleides. It thus appears to us that the ground band moments of inertia are around 30 % of rigid, rather than = 70 %, and the deformations @ are = 0.34, not the 0.55 of Zicha et al. The question now is what moment of inertia should have been used in the fission angular momentum calculations of table 1. According to eqs. (11) and (14) the average angular momentum is approximately proportional to the one-fourth power of the moment of inertia. Thus, if we should use a moment of inertia 30 % of rigid, we would get a lowering of &, by about 25 % to values of around 4.5. On the other hand, it is more Iikefy that around the scission barrier the fragments are not “cold” but have sufficient excitation of internal degrees of freedom that the moment of inertia would approach the rigid value, in closer accord with the original numerical estimates of table 1.

Coulomb excitation may excite downward in spin as well as upward. Thus, the Coulomb excitation correction will not add arithmetically to the average I-value at scission but may add in a random-walk fashion. Thus in the static quadrupole case, the final average i-value may go to around 8. In the case of a quadrupole oscillation critically damped to spherical shape, the average I-value may go to around 6.

478

1. 0. RASMWSSEN et al.

The kinetic energy distribution has a considerable spread about 180 MeV. The lower kinetic energy events imply larger values of aZ aad Ed. For the numeric& parameters here, it is a fair a~~ro~rnat~~~ to drop aif but the leading term for K in eq. (4). We then have

We cannot eliminate the deformations without specifying & ratio of CQto a3 e However, it is clear that the depe~denoe of 1y on the kinetic energy wilf be rather weak, since dfie kinetic energy appears in the numerator and d~~~~nator_ We shall not attempt here to generalize the treatment to the case of non-zero total angular momentum or to both fragments deformed. The latter case is not as formidable as might be supposed for coupling three angular momenta (two rotational and one orbital). The problem is simpliftcd since the orbital moment of inertia is so much larger than the rotational rnorne~ts~ and since the angles of bending or ~gg~~ng are smail. Preliminary calculations show that the problem can be approximated as the solution of two coupled two-dimensional harmonic oscillators. Let us Anally discuss case (a) of the fission mechanism with thermal equilibrium of collective degrees of freedom at the scissian valley. The theory developed in sect, 3 remains almost completely valid. Indeed, our derivation of the bending restoring force constant klin eq. (4) holds equal& weff for the ~&fey or the barrier top, since there is a balance between the slopes of the electrostatic and the nuclear potential at both points. The numerical calculation for the angular momentum distribution in the scission valley would proceed just as that given in table 1 except that somewhat sm&er values of C-X and of distortion parameters a2 and g3 would enter eq. (41, and one should explicitly substitute the ~on~omb energy at the valley where Ekin appears. The final answer for la7 would not be much different from those of taMe 1, since changes of various factors will tend to compensate. The thermal equilibrium at the valley is characterized by a collective temperature which might be large enough for the excited states of the bending mode to be probable. As is seen from fig- 4, these excited states include somewhat larger spin w&es than the grctund state, 6, Comparison with experiment There have been several papers that have related experimental data to tha primary distribution of spins in f&&n fragments. Strutinsky 16) in 1959 pointed out that the an~so~~~~~of fragment gamma radiation with respect to the fission direction could be ~~a~~t~~~~~~~ understood by assuming a substanthf portion of electric qnad~po~e radiation and the predominance of fragment initial angular momenta perpendicular

FISSION

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FRAGMENTS

to the fission direction. He roughly estimated the angular momentum that might be imparted by post-scission Coulomb excitation to be 1,” x 20 . Ramanna et al. ““) in 1961 published measurements of the angular correlation of prompt neutrons and fission fragments. They discussed how the spin distribution may affect the observation, but they did not attempt to revise Strutinsky’s estimate16). Croall and Willis “) in 1963 published studies of relative yield of *lSe and 83Se isomeric pairs from thermal neutron fission of 23gPu. Applying the statistical method of Vandenbosch and Huizenga “), they assumed a distribution of N(J) = (25 + 1) exp [_

Jy]

and inferred that the r.m.s. spin value (T)* = bJ2 is about 8 for these selenium fragments. Sarantites, Gordon and Coryell 2g) later published isomer-yield studies on 13’Te and 133Te from thermal neutron fission of 235U. They infer r.m.s. spin values of 6.0 + 1.5 and 5.9 + 1.5 for the two isomeric pairs, respectively. Hoffman I’) in 1964 published his data on anisotropies of the integrated gamma spectrum above 250 keV foIlowing thermal neutron-induced fission of 23pPu, 233U and z3SU. He states from his analysis that the fragments “appear to have 6 to 8 units of angular momentum oriented preferentially about 90” to the fission axis after the emission of fission neutrons”. We regard this analysis as subject to more uncertainties than the isomer-yield work, since assumptions must be made about the spin sequence and the multipolarities of countless gamma cascade paths contributing to the unresolved gamma spectrum. Thomas and Grover 30) in 1967 presented a theoretical analysis of the average energy carried off by gamma rays following fission. They showed that the initial spin distribution has a significant influence on the average energy. From review of the above-mentioned studies, they chose to use an r.m.s. spin value of 8.4 in their calculations, and they achieved satisfactory agreement with experiment. Referring to our table 1, we see that the r.m.s. spin value we predicted at scission for lo8Ru is about 7 in reasonable agreement with the isomer-yield measurements, though they are unfortunately taken in different mass regions. Of course, our theory is not directly tested by experiment, but it seems likely that post-scission Coulomb excitation plays only a minor role in altering the spin distribution fixed by the zeropoint motion in the bending mode at scission. One of us (J.O.R.) wishes to express thanks for the hospitality in summer 1968 of the Technische Hochschule, Miinchen, where this work was begun, and to the Niels Bohr Institute where it was further developed. At the latter institution, we had the valuable opportunity for discussions with V. M. Strutinsky. Thanks are due to Professor Marshall Fixman, Yale University, for pointing out that our wave equation is the angular wave equation occurring in ablate spheroidal coordinates.

480

J. 0.

RASMUSSEN et a[.

Partial support of the U.S. Atomic Energy Commission through contracts 34 (Berkeley) and AT(30-1) 3909 (Yale) is gratefully acknowledged.

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