Asymmetry in Nuclear Fission

1518 C. H. BINGHAM AND G. T. FABIAN

IO.O

I,O

E

O

OIJJM

o olO

LLJ

CLLLI

Lt

C5

O.OI

/1 m/

Xa

96Zr (d, p)

Ed=33.3 MeV

1,108 MeV—x 0.5

~ 2=2

~ ~ 1.399 MeV

'L

l

\

00 MeVx 0.048=0

IO.O

~—=, 2.265 MeV~ 2=5Z0

O 4LLJ(A 8

M

CLC3

I— a$IJJKUJ

C5

3.731 MeV-X=5

X. ~

— 1.265 MeV:x 0.034=4

O.OI

o~

E L

O.OOIIO 20 30 40 50

CENTER-OF-MASS ANGLE (deg)

60 O.OOIIO 20 30 40 50

CENTER-OF-MASS ANGLE (deg)

60

FIG. 11. Experimental angular distributions for l =0and l =2 levels in 96Zr(d, p) (points) in comparison withdistorted-wave calculations (smooth curves).

FIG. 12 ~ Experimental angular distributions for l =4and l =5 transitions in 9~Zr(d, p) (points) in comparisonwith distorted-wave calculations (smooth curves).

the small isotopic abundance of the target andhence the results are only approximations to thecomplete picture.

The spectroscopic strengths for the 2d„„3s|i„and 1g„, states agree well with the expected values.The measured spectroscopic strength for the 2d„,states is somewhat higher than expected and thatfor 1h»» states is -50-60% of that expected.Similar results were reported previously for "Zr

and 92Zr. '

ACKNOW( LEDGMENTS

We are indebted to M. L. Halbert and J. B. Ballfor many valuable discussions and to M. L. Hal-bert for his assistance in taking the data. Thecareful plate scanning of Phyllis Wagner and Bar-bara Monroe is gratefully acknowledged.

*Research supported by the Army Research Office—Durham under a grant to The University of Tennesseeand by U. S. Atomic Energy Commission under contractwith Union Carbide Corporation,

)Present address: Air Force Weapons Laboratory,Kirtland Air Force Base, Albuquerque, New Mexico87117.

~C. R. Bingham, M. L. Halbert, and R. H. Bassel,Phys. Rev. 148, 1174 (1966).

2B. L. Cohen and O. V. Chubinsky, Phys. Rev. 131,2184 (1963).

3C. E. Brient, E. L. Hudspeth, E. M. Bernstein, andW. R. Smith, Phys. Rev. 148, 1221 (1966).

4R. L. Preston, H. J. Martin, and M. B. Sampson,Phys. Rev. 121, 1741 (1961).

5J. K. Dickens and E. Eichler, Nucl. Phys. A101, 408(1967).

J. S. Forster, L. L. Green, N. W. Henderson, J. L.Hutton, G. D. Jones, J. F. Sharpey-Schafer, A. G. Craig,and G. A. Stephens, Nucl. Phys. A101, 113 (1967).

~G. Simmerstad, M. Iverson, and J. J. Kraushaar,Technical Progress Report, University of Colorado Nu-

NEUTRON SHELL STRUCTURE IN "Zr, "Zr, AND "Zr. . . 1519

clear Physics Laboratory, Report No. Coo-535-603,1969 (unpublished).

C. R, Bingham and M. L. Halbert, Phys. Bev. C 2,2297 (1970).

9N. Baron, C. L. Fink, P. B. Christensen, J. Nickles,and T. Torsteinsen, Bull. Am. Phys. Soc. 17, 50 (1972).

E. Newman, L. C. Becker, B.M. Preedom, and J. C.Hiebert, Nucl. Phys. A100, 225 (1967).

~~M. P. Fricke, E. E. Gross, B, J. Morton, andA. Zucker, Phys. Rev. 156, 1207 (1967).

W, Whaling, in IJandbuch der physik, edited byS. Flugge (Springer-Verlag, Berlin, Germany, 1958),Vol. 34, pp. 193-217.

~~Program developed by R. M. Drisko, R. H. Bassel,

and G. B.Satchler. .~4G. B. Satchler, private communication.

C. B. Bingham and M. L. Halbert, Phys. Bev. 158,1085 (1967).~~F. G. Percy and A. M. Saruis, Nucl. Phys. 70, 225

(1965).~YJ. K. Dickens, R. M. Drisko, F. G. Percy, and G. B.

Satchler, Phys. Letters 15, 337 (1965).C. B. Bingham, M. L. Halbert, and A. R. Quinton,

Phys. Bev. 180, 1197 (1969).~ I. Talmi, Phys. Rev. 126, 2116 (1962).J. B. Ball and C. B. Fulmer, Phys. Bev. 172, 1199

(1968).

PHYSICAL REVIE W C VOLUME 7, NUMBER 4

Asymmetry in Nuclear Fission

M. G. Mustafa*Oak Ridge National Laboratory, f Oak Ridge, Tennessee 87830

A PRIL 1973

and

U. Moselle.University of Washington, 5 Department of Physics, Seattle, Washington 98295,

and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

and

H. W. SchmittOak Ridge National Laboratory, Oak Ridge, Tennessee 87830

{Received 8 September 1972)

The two-center shell model for fission has been generalized to include asymmetric defor-mations. The calculation of the potential energy involves four independent shape variables,where only two were required in the symmetric calculations. Potential energy calculationshave been carried out for Pb ~ Po 3 U 4 Cm Fm Fm, and Fm. Asymmetricfission is found to be energetically preferred in BU', 24 Cm, and 5 Fm; and symmetric fis-sion is preferred in Pb, 2 Po, Fm, and Fm. Two of these nuclei, namely ~U and

Po, have been studied in detail. It is seen that the asymmetry in S~U remains almost con-stant from the second saddle to scission, whereas in Po (and also in Pb), the preferredshape changes from asymmetry in the region of the second saddle to symmetry in the regionof scission. The results for Fm isotopes indicate that there is a transition from asymmetricfission in the lighter Fm isotopes to symmetric fission in the heavier Fm isotopes. Thepreference for symmetric mass division in 2 Fm is very strong, since two double-magic

&OSn82 fragments are formed at symmetry. In general, the structures which appear in thepotential energy surfaces are the results of an interplay between compound-nucleus shellstructure, fragment shell structures, and liquid-drop-model energies. Comparisons of ourresults with experimental observations indicate that the observed mass distribution is corre-lated with the potential energy surface in the neighborhood of scission.

I. INTRODUCTION

The most significant achievement in the calcula-tion of potential energy surfaces for nuclear fis-sion and heavy-ion reactions in recent years hasbeen the development of methods for calculatingshell corrections to smooth liquid-drop-model

surfaces. ' These methods allow one to calculatepotential energy surfaces' ' rather quantitatively,and thus help one to understand nuclear fissionand fusion in a more detailed way.

In particular, it has been a long-standing hopethat such potential energy calculations would per-haps provide a basis for the observed dominance

1520 MUSTAFA, MOSEI, AND SCHMITT

of asymmetric mass divisions in low-energy fis-sion of the actinides. Although this mass asym-metry has long been considered to be a conse-quence of shell effects, it is only recently thattheoretical calculations, by Moiler and Nilsson,indeed established an instability of the secondbarrier to asymmetric deformations. It is ouraim in this paper to show the physical reason forthis instability and to show its connection withspecific fragment structure properties.

Since all calculations based on extended Nilssonmodels, i.e., one-center single-particle poten-tials, are not suitable for an investigation of thisproblem, we have undertaken a systematic studyof potential energy surfaces for nuclear fissionby generalizing the two-center shell model'' toasymmetric deformations. ' Since this general-ized two-center model can describe the correctshell structure of a fissioning nucleus as well asthat of two noninteracting, well separated nuclei,it is very suitable for an investigation of the fis-sion problem. In the next section we will, there-fore, present the mathematical details of themodel and discuss some of its properties. Insubsequent sections we will systematically investi-gate the dependence of the nuclear potential en-ergy on asymmetric deformations and the connec-tion between the topology of the potential energysurface and the structure of the nascent frag-ments.

II. ASYMMETRIC TWO-CENTER MODEL

The Hamiltonian used for the calculation ofsingle-particle energies appropriate for highlydeformed, and even separated nuclear shapes,as they appear in nuclear fission and heavy-ioninteractions, is that of the asymmetric two-center model. Since earlier versions of thismodel either violated nuclear saturation prop-erties' or were too schematic for a detailed quan-titative investigation of fission potential energysurfaces, ' we introduce a new generalization ofthe symmetric two-center model to asymmetricshapes, which avoids these difficulties.

We make the following ansatz for the single-particle Hamiltonian:

H = T + —,' m &u~,. '(z)p'+ —', m ~„.'(z —z,.)'+ V„„+V(l, s)

symmetric calculations, ' by choosing:

., '~;( — .)'8(l I-I, I),where 8(x) =0 for x&0 and t)(x) = 1 for x&0.

It should be mentioned here that this potential(or shape) parametrization is suitable both forthe description of heavy-ion interactions and ofnuclear fission. In the former process the twoions approach each other only slightly distortedcompared with the fission fragment deformations.This case is contained in the present parametri-zation by setting $,. =0. The shapes describableby the potential are then those of two unequalseparated or overlapping spheroids, i.e., thoseappearing before and about the contact point inheavy-ion reactions.

The function ~~,.(z) in Eq. (l) is chosen to de-pend on z in the same way as the purely z-de-pendent part of the potential:

(u„. '(z) =('up, . '+n, Hz —z,)'+(, (z —z,.)'] 9(IzI —Iz; I),

The quantities ~~;, o!, , $,. are defined in AppendixA R

p z and ~p, are inverse ly proportional to the

60

50

40

—50

20

'IO

with i =1 for z & 0 and i =2 for z & 0. The nuclearshapes that can be described by this model areessentially those of two nonidentical semispher-oids connected by a smooth neck. The centers ofthe semispheriods are located at a, and z, . Thesmoothing of the neck is achieved, in direct gen-eralization of the corresponding expression in

FIG. 1. The lower portion of the figure shows a nucle-ar shape described by the Hamiltonian given in Eq. (1);the upper portion shows the corresponding z-dependentpart of the single-particle potential and is given by V(z)=2m~«'(z-z;) +m~„g;(z —z;) t)(Iz I

—Iz;I) w'th i=1forz &0 and i=2 for z &0 (see Sec. II and Appendix A).It is to be noted that this potential is independent of thetransverse semiaxes a& and a2.

ASYMME TRY IN NUC L EAR FISSION

transverse semiaxes a, and a, (see Fig. 1), and

o, a.nd $,. are chosen to allow a smooth transi-tion from (up, to (op, .

The nuclear shape corresponding to the single-particle potential V(p, z) in Eq. (1) is obtained inthe usual way by assuming that the nuclear sur-face follows an equipotential surface with thevalue V,:

V( p, z) = Vo p = p(z) ~ (4)

For the potential value V, we take, as usual, V,= 2m&0 8 with A being the nuclear radius and (do

the oscillator frequency given by @x,= 41A '"MeV,where A is the nuclear mass number. The shapeparameters can then be determined from variousconditions for smooth transitions of the potentialand its derivatives at z =0, volume conservation,and the condition that for large separations (largez, —z,) two independent Nilsson potentials shouldbe contained in the present parametrization. (Fordetails of these conditions see Appendix A). Theserequirements finally lead to four independentshape parameters for which we choose the neckradius D (in fermis), the volume ratio of the frag-ments A. , and the two additional coordinates n

(describing the relative maximum p dimensionsof the fragments) and o (describing the elongationof one of the two fragments). The exact definitionsfor these quantities are given in Appendix A. (Seealso Fig. 1).

The angular momentum-dependent term V(1, s)is taken to be:

C(z, )1, ~ s +D(z, )(1,' —(1,')),C(z, )1, s+D(z, )(1,' —(T,')),

where l, and l, are the pseudoangular momentawith respect to the two centers at z, and z, . Thesepseudoangular momenta are defined in terms ofthe stretched coordinates appropriate to each ofthe two half spheroids. ' In order to assure acorrect transition to the appropriate C and Dvalues for the fragment regions, we have intro-duced the Nilsson parameters I(. and p, in the fol-lowing way:

C(z,.) = -2h&V(z;)~(z;),

D(z,.) =-', C(z,.)p, (z,.),where 2 is defined by:

polation variable is (d. The values of g and p, aregiven in Table I and are based on the values ofNilsson et al. ' and Arseniev, Sobiczewski, andSoloviev. "

The Hamiltonian (1) is then diagonalized in abasis consisting of eigenstates of the basis Hamil-tonian:

Ho = T + 2 I (0& p + 4 m (d (z —z )

Here (dp is the arithmetic average of (dp] and (4p2.The basis functions of II, can be obtained by asimple matching procedure. ' This basis choice,compared with the standard deformed harmonic-oscillator basis used by most other groups, hasthe advantage that it already contains the over-alldeformation of the nuclear shape, even at scis-sion and beyond. This means that the most im-portant deformation-dependent terms in the Hamil-tonian are already diagonal and only small pertur-bations have to be diagonalized; thus good con-vergence is assured.

In contrast, we note that the one-center de-formed harmonic-oscillator basis leads to a pro-gressively worse convergence when applied to thedescription of shapes at the scission configura-tion and beyond. This precludes any descriptionof heavy-ion interactions and of the latter stagesof fission in which such configurations occur.The two-center basis states have none of theseshortcomings, and the convergence even improveswith increasing separation.

This basis choice also leads to a natural general-ization of the term (1,. ') which in the standardNilsson model is given by" N(N+3) j2. Here N isthe principal oscillator quantum number. Sincethe eigenenergies of B, are given by'

E, = (N, + 1)5 (up

+ (n„+-', )I(u„. .

with n„. being an integer only in the cases of either

TABLE I. The table lists the single-particle parame-ters f( and p, for protons and neutrons for the two com-pound-nucleus regions discussed in the text and for thefragment regions. They are, except for very smallchanges, taken from Nilsson et al, . (Ref. 2) for the twocompound-nucleus regions and from Arseniev, Sobic-zewski, and Soloviev (Ref. 10) for the f'ragment regions.

(d(z;) = ((d ' (d )

This assures a continuous transition from theoscillator constant for the fissioning nucleus tothe two different oscillator constants for the finalfragments. The quantities ~(z,.) and g(z,.) areinterpolated linearly between the values for thecompound and the fragment regions. The inter-

Mass region

A ~100A ~132A =200A 250

Protons

0.0688 0.5580.0671 0.5720.0610 0.6260.0580 0,645

NeutronsK p,

0.0638 0.4910.0638 0.4930.0636 0.3700.0635 0.330

M USTA FA, NOSE L, AND SC HMITT

zero or very large separations, we set

(1,.2) =2(N +n„.)(Np+n„+3),. (10)

thus assuring the correct transition of this termfrom its proper value in the fissioning nucleusto that in the two fragments (see, e.g. , Fig. 2 ofRef. 8).

The actual diagonalization of (1) has been per-formed in the basis mentioned above. We foundthat taking the 300 lowest eigenstates of H, (in-stead of using a specified number of shells) ateach deformation assures good convergence—better than 10 keV in the shell corrections.

The choice of oscillator potentials in (1) is nota necessary choice, since the two-center modelcan be formulated with other potentials as well,e.g. , Woods-Saxon potentials. It has been claimed,especially by Bolsterli et al. ,

' that the harmonic-oscillator potential is inferior to the more real-istic finite depth potentials. This claim, althoughcorrect for the description of many physical phe-nomena (e.g. , single-particle reactions), is mis-leading in the present context. First, the asymp-totic behavior of bound-state wave functions,which is different in the two models, nowhereenters into the calculation of potential energysurfaces. Second, the seemingly obvious advan-tage of using a more realistic finite depth poten-tial is lost because the Strutinsky shell correc-tion method requires the presence of single-parti-cle states rather high (=20 MeV) above the Fermisurface. This necessitates the discretization ofthe continuum into physically meaningless single-particle states that appear as a byproduct in diag-onalization of the finite depth potentials in oscil-lator basis functions. In other words, the use ofthe Strutinsky shell correction method necessi-tates, in effect, the introduction of infinitely highpotential walls outside the nuclear radius, evenin the case of "finite depth" potentials. Thus, theonly remaining advantage in the use of finite depthpotentials is the better description of the surfaceregion, which makes the use of the l' term un-necessary.

After having obtained the single-particle energiesof (1) as described above, shell corections 5U

are generated from these by using t»e Strutinskyprescription' with a smearing width of 1.2@~ anda correction polynomial of sixth order.

The liquid-drop model has been calculated forthe shapes given by Eq. (4) with the parametersused by Myers and Swiatecki, ' except that the sur-face symmetry coefficient I(., has been changedto the value given by Seeger. " This change wassuggested by Johansson, Nilsson, and Szymanski"who obtained second barriers which were system-atically too high in the actinide region when the

original Myers and Swiatecki value was used. A

similar value of I(:, has also been suggested by

Pauli and Ledergerber. ' The most realistic valuefor z, , of course, can only be obtained in a simul-taneous refit of the liquid-drop-model plus shell

1.5—

2.0—

2.5

03.0—

V) 3.5—a 2

E: 340—

LLI' 4

5.0—

6.0— 2::3:236

U3:2

J~ l I J78 88 98 108 118 128 138158 148 138 128 118 108 98

FRAGMENT MASSES (aralu)

148 15888 78

FIG. 2. The two-dimensional potential energy surfaceV(X,D) for ~36U is shown A. is the volume (mass) ratioof the portions of the nucleus on either side of. the neckplane, and D is the radius of the neck in fermis. Theenergy has been mini, mixed with respect to n and 0,which are the other two shape parameters describingthe asymmetric nuclear shape (see Appendix A). Thezero reference energy is at the ground state of 23~U.

The numbers labeling th. e contour lines give the contourenergi. es in MeV. The solid lines are the results of ourcalculations; the dashed lines have not been calculated,but are included as approximate curves for the complete-ness of the V(A, ,D) plot.

AS YMME T RY IN N UC L EAR F IS SION 1523

and those in the A-200 region are

G~ = [20.8 —0.0S33(A —180)]/A MeV,

G„=I14.5 —0.0233(A —180)j/A MeV.

(12)

They are taken to be independent of deformation,since recent calculations in the range 180&A. ~ 212indicate that the assumption of a surface-dependentpairing strength is incompatible with experimentaldata on fission barriers in that mass region. 'The pairing energy is then given by

f g2Z, =Q 2+a, v, ' —2+e, ——

P, n u P(13)

The total potential energy is given by

t/'=ELM +Ep +5U,

corrections to experimental masses, a calcula-tion not yet undertaken. The results for massasymmetry in fission and the indications of frag-ment shell structure effects in the potential ener-gy surfaces that we shall present in this paperare not sensitively dependent on ~, . The quantityg, is, however, more critical in detailed calcula-tions of barrier heights.

The pairing interaction has been taken into ac-count in the BCS formalism using Z levels for theprotons and N levels for the neutrons. The pairingstrengths that give good fits to the odd-even massdifference in the heavy-element region are

Gp = MeV, G„= MeV,19.4 13.8

where E~D„ is the liquid-drop-model energy.Our calculations have been carried out by fixing

successive values of D and A., and then performingfor each of these a two-dimensional potential en-ergy surface calculation. By thus doing a fullfour-dimensional calculation, a correct deter-mination of the saddle-point location within thechosen shape parametrization is assured. "

III. RESULTS

Potential energies have been calculated for thefollowing fissioning nuclei: '"Pb, '"Po, '"U,

cases, namely '"U and '"Po, have been studiedin detail.

Our procedure for carrying out the calculations,in general, was to fix values of the volume (ormass) ratio & and the neck thickness D, and thento minimize the potential energy with respect ton and o. %e thereby obtain a matrix of valuesV(A, D), for a given fissioning nucleus. The elimi-nation of the two other shape parameters by mini-mization is not meant to imply that the fissioningsystem always follows the minimum energy path;rather, this procedure just permits the displayof our results in terms of the two most importantfission variables ~ and D. The quantity D mayserve as an approximate fission coordinate in theregion beyond the second saddle point, where the

20

44lX

236U

——) =1.0 (N—-—) =1.27(N

15—

10—E

5—Cl /:~/:i

-F

iPaayg

~L

I' J//

-67.5

----- X=1.70 (NH/Q' = &49/87)------ &=1.45(e /u = 140/96)H/ L

X=1.50(NH/NL = 142/94)

I I I I I I I

7.0 6.5 6.0 5.5 5.0o (fm)

4.5 4.0 3.5

FIG. 3. The potential energy V(D) as a function ofneck thickness is shown for selected values of A, , for236U. These curves represent selected vertical cutsthrough Fig. 2. The energy has been minimized withrespect to n and a, and normalized to zero at theground state. Note the rather sudden change from sym-metric to asymmetric shapes at D —=5.6 fm.

-10 I

-20 -15I

-10I

0 5LENGTH (fm)

I

10I

15

FIG. 4. Shapes of the 3 U nucleus at various pointsalong the minimum potential energy path. The shapeslabeled A, B, and C are reQection symmetric (A, =1)and correspond to the neck radius 7.05 fm (approximateground state), 6.4 fm (first saddle), and 5.8 fm (secondminimum), respectively. The shapes labeled D, E, and

F are asymmetric and correspond to neck radius and vol-ume ratio of 5.0 fm, 1.5 (second saddle), 2.5 fm, 1.45,and 0.5 fm, 1.45, respectively.

MUSTAFA, MOSEL, AND SCHMITT

constriction degree of freedom becomes impor-tant. '

23692 144

The two-dimensional potential energy surfaceV(A., D) for '"U is shown in Fig. 2. The groundstate of this nucleus occurs at D = V. 05 fm, cor-responding to a spheroidal shape. " Following theminimum potential path to scission we pass overthe first barrier at D =6.4 fm, for which the nu-clear shape is reflection symmetric. Continuinginto the second minimum at D =5.8 fm, we findthat a reflection-symmetric shape is still pre-ferred, but that the potential becomes very softtowards asymmetric deformations. The mini-mum energy path crosses the second barrier atD =-5.0 fm with a quite asymmetric shape, namelyA.:—145/91. Continuing towards scission, i.e. ,towards smaller D values, we see that the rnini-mum in the potential energy surface shifts slightly,to A. =140/96, in excellent agreement with the mostprobable mass division observed in the low-energyfission of '"U."

We note also, from this figure, that the asym-metric barrier is favored by about 2.3 MeV rela-tive to the symmetric barrier, and that near scis-sion the most probable asymmetric shapes arepreferred by -7 MeV relative to symmetry. Thecal.culated barrier height (saddle-point energy)is in satisfactory agreement with the experimentalvalue. "

In Fig. 3 we have plotted a few vertical cutsthrough Fig. 2. That is, we plot V(D) for selected

values of X. The usual double hump in the barrieris apparent, and differences for different ~ valuesare shown. This figure does not imply that theoccurrence of a particular final A. at scission cor-responds to a process that has followed one ofthese curves; rather, the system may move overthe full four-dimensional potential surface, andany final A. may result from events, for example,which cross the saddle at A. =145/91 (or any otherA), depending on the quantum dynamics of the pro-cess.

The shapes of the nucleus at various points alongthe minimum potential energy path are shown inFig. 4. The shapes A, 8, and C are reflectionsymmetric; shapes D, E, and F are asymmetric.It is interesting that near scission the heavy frag-ment appears to have a more nearly sphericalshape than the light fragment.

In Fig. 5 we show a comparison of saddle-pointshapes for different values of ~. The point to notehere is that the shapes are not very different, andthe nucleus does not have to change its shape verymuch to cover the full range of A, .

A similar, and perhaps more important com-parison is shown in Fig. 6 for shapes nearerscission, at D =2.5 fm. Again it is clear thatthe full range of A. is accessible with relativelysmall changes in shape.

Since the calculations of the full four-dimension-al potential energy surfaces are very time con-suming, our calculations for most other nucleiare restricted to cuts V(X) for different values ofD around and beyond the second saddle point, aslocated within the symmetric calculations. To

E4

(A

C5

0

I I I I I I I I I I I

236U A B20

15

10

E

2v) 5

CL 0—

2&6U

B

C— A

—16 —12 —8 —4 0 4 8 12LENGTH (fm)

16-10

—20 -15 -10I

0 5LENGTH (fm)

10 15 20

FlG. 5. Shapes of the ~U nucleus for different valuesof A. are compared at D =5.0 fm, corresponding to thesecond saddle region. The shapes A, 8, and C corre-spond to A, =1.0, 1.45, and 1.7, respectively.

FIG. 6. Shapes of the 6U nucleus for different valuesof A, are compared at D =2.5 fm, corresponding to theapproach to scission. The shapes A, 3, and C corre-spond to A. = 1.45, 1.0, and 1.7, respectively.

ASYMMETRY IN NUCLEAR FISSION

illustrate the relationship between the full poten-tial energy surface and these cuts we show inFig. 7 the curves V(X) for "'U. The curve cor-responding to the neighborhood of the second sad-dle point is labeled D =5.0. This curve gives agood indication of the behavior of the potentialenergy surface about the saddle, although it alonedoes not suffice to localize the saddle point un-equivocally.

Similarly, a key indication of the behavior ofthe potential energy surface near scission is thefunction V(A. ) for D =D „,where D „is somevalue of D approaching scission, chosen generallyin the range 2 to 4 fm. Such a curve for '"U islabeled D =2.5 in Fig. 7.

The remaining curves in Fig. 7 correspond toD = 5.7, close to the second minimum in the poten-tial energy surface, and to an intermediate pointbetween D = 5.0 (saddle point) and D = 2.5, namelyD =4.5. It is seen in Fig. 7, as in Fig. 2, thatasymmetric shapes (A. =145/91) are favored atthe saddle, and that the potential energy minimumshifts slightly, to X=140/96, as D decreases to-ward scission.

130118

I

135113

I

NH

N

140108

I

145103

15098

I

10

Let us discuss one further aspect of the "'Uresults. It has long been known that the Q valuesfor fission, i.e., the available energy for fissioninto various fragment mass divisions, favor divi-sion into final fragments with A. =132/104. Theminimum potential energy near scission is seenhere, however, to occur at A. =140/96, in agree-ment with experiment. Our explanation for thismay be stated approximately as follows: When A.

is precisely (or very close to) 132/104, the double-

magic mass-132 partner tends toward sphericityto take advantage of its large binding energy. In

so doing, however, the repulsive Coulomb poten-

120 125 150 155116 111 106 101

I I I

bfH

hfL

14096

I

14591

I

15086

I

2x

C9LL

0IJJ

2

0bJ

-2

-8 ——1.0

I

1.2 1.3 1.4 1.5), VOLUME RATIO

I

1.6 1.7- 1.8-10

1.0I I

1.2 1.3 1.4X, VOLUME RATIO

1.5

FIG. 7. The potential energy for 23~U as a function ofA, , the volume ratio of the portions of the nucleus oneither side of the neck plane, for several values of D,the radius of the neck in fermis. The energy has beenminimized with respect to n and 0 and normalized tozero for the ground-state energy.

FIG. 8. The potential energy for 4 Cm as a functionof A. , the volume ratio of the portions of the nucleus oneither side of the neck plane, for several values of D,the radius of the neck in fermis. The energy has beenminimized with respect to G.' and cr and normalized tozero for the ground-state energy.

1526 MUSTAFA, MOSE L, AND SCHMITT

tial tends toward higher values, so that the totalpotential energy is actually higher than that atX =140/96, where the additional -8 nucleons inthe heavy fragment apparently provide an optimumcompromise between the increased binding energyof nuclei approaching double-magic configurations,and the reduced stiffnesses of nuclei receding fromclosed-shell configurations.

248a. „Cm»,

The potential energy for this nucleus has beenstudied by examining cuts in the V(A., D) surface,showing V(X) at D =5.0, 4.0, and 3.0 fm. The re-sults of these calculations are shown in Fig. 8.

The symmetric saddle point for this nucleusoccurs at 4.0 ~D ~ 5.0 fm; the curve labeled D=5.0 in Fig. B,indicates that the preferred shapeis reflection symmetric. As the system continuestoward scission, we see that asymmetric shapesbecome preferred (see curve for D = 4.0) and thispreference becomes even stronger at D =3.0, asevidenced by the stronger minimum at A. =144j104.

252 258 264FC. 100Fm152' 100Fm»8, » 100 164

The potential energies for these nuclei havebeen studied„as in the case of '"Cm, by examin-ing cuts through the V(A., D) surfaces, showing

V(X) for several values of D. The results areshown in Figs. 9-11.

The saddle point in '"Fm, from reflection-sym-metric calculations, ' occurs at about D =4.87.The curve showing V(A) for D =4.87 in Fig. 9 indi-cates that here the preferred shape is still reflec-tion symmetric. Similar results are obtained for' 'Fm and '~Fm, as shown by the curves labeledD = 5.0 in Figs. 10 and 11.

As D becomes smaller, in the descent towardsscission, we find that the minimum potential en-ergy in '~Fm shifts rather quickly to an asym-metric shape (curve labeled D =4.4 in Fig. 9). Theasymmetric shape then appears to remain pre-ferred through intermediate stages (curve labeled

NH

Nl

130122

135117

NH

Nl

'I 40112

130128

135

123I

'I40

118

I

145

113

150

108

I

0

el) 4 .0 &- -8KUJZ,'ILJ

0=4.0

-20 D=2.0

-20&.0

I

1.2

X, VOLUME RATIO

1.0 1.2

X, VOlUME RATIO

1.3

FlG. 9. The potential energy for 5 Fm as a functionof A, , the volume ratio of the portions of the nucleus oneither side of the neck plane, for several values of D,the radius of the neck in fermis. The energy has beenminimized with respect to n and cr and normalized tozero for the ground-state energy.

FIG. 10. The potential energy for ~ Fm as a functionof A, , the volume ratio of the portions of the nucleus oneither side of the neck plane, for several values of D,the radius of the neck in fermis. The energy has beenminimized vrith respect to m and cr and normalized tozero for the ground-state energy.


D = 4.0) and onward towards scission (curve D = 2.0).In contrast, the preferred shapes remain sym-

metric throughout the descent to scission for '"Fmand '"F, as seen in Figs. 10 and 11. The prefer-ence is quite strong, i.e., the potential minimumis quite sharp in '"Fm, where symmetric fissionresults in the formation of two double-magic132„Sn„nuclei. The potential near scission in '"Fmis softer towards asymmetry, exhibiting a some-what Qat shape as might be expected in the transi-tion region between asymmetry ('"Fm) and sym-metry ('"Fm).

Measurements of mass distributions in the fis-sion of Fermium isotopes show asymmetric peaksin '"Fm "and '"Fm,"and a symmetric peak in'"Fm." The peak-to-valley ratio for '"Fm is,however, only 12, a small value relative to theusual peak-to-valley ratios for the low-energy

2.0—

3.0—

//trI

/ I

!I

fission of actinides. A broader peak, with somedip in yield at symmetry has been found for'"Fm."'" The results of our calculations arequalitatively in accord with the measurements.The most probable heavy-fragment mass for '"Fmis calculated to be 140 amu, as observed for "'Fmand Fm. Although the rather flat potential cal-culated for Fm would indicate a broader, moreflat-topped mass distribution than is observed,there are two possible reasons why such a dif-ference might occur; (l) The calculations arenot refined sufficiently to give complete details

340f24

&45

119

150114

3.5—/

4.0 -I

//» tIrn

5.0 —) )~16

14

~.zo.s «.~ ~o ~ zo.s

I ' x /

pp e ~o'«. r/ /

2ii is

6.0

6.5—

7.5210p

Oe

-20

8.0—65145

I I l l .75 85 95 105 115 125 135 145135 125 115 105 95 85 75 65

FRAGMENT MASSES (omu)

-28).0 1.2

X, VOLUME RATIO

1.3 1.4

FIG. 11. The potential energy for ~~4Fm as a function

of A, , the volume ratio of the portions of the nucleus on

either side of the neck plane, for several values of D,the radius of the neck in fermis. The energy has beenminimized with respect to n and a and normalized tozero for the ground-state energy.

FIG. 12. The two-dimensional potentM energy surface

V(A, ,D) for po is shown; A, is the volume (mass) ratioof the portions of the nucleus on either side of the neck

plane, and D is the radius of the neck in fermis. The

energy has been minimized with respect to o. and 0,which are the other two shape parameters describingthe asymmetric nuclear shape (see Appendix A). The

zero reference energy is at the ground state of Po.The numbers labeling the contour lines give the contour

energies in MeV. The heavy lines are the results ofour calculations; the thin dashed lines have not been

calcu1ated, but are included as approximate curves forthe completeness of the VP. ,D) plot.

1528 M USTA FA, MOS E I, AND SC HMITT

of the distributions, and there may be significantdynamic effects not yet taken into account, and

(2) the measurements were made by bombarding'"Fm with thermal neutrons, resulting in an ex-citation energy of -6 MeV in the fissioning nucleus.The effect of increased excitation energy on thefargment mass distribution in this specific caseis not known, but acts generally to enhance thesymmetric yield. A measurement of the massdistribution for the spontaneous fission of Fm,which is expected to be rather broad according toour calculations, would therefore be very impor-tant for a better understanding of this transitionregion.

The important point here is that a transitionfrom asymmetric fission in the lighter Fm iso-topes to symmetric fission in the heavier Fm iso-topes occurs and is accounted for in the calculationsof potential surfaces, in which fragment structureeffects are evident. Specifically, symmetric fis-sion becomes prominent when the number of neu-trons reaches a value, apparently N-158, at whichtwo nearly-double-magic fragments can be formed.

21084 126

Some years ago, in the liquid-drop-model studiesof Nix and Swiatecki, "it appeared that the liquid-drop model would successfully describe the fissionof nuclei lighter than about radium. These nuclei,at medium excitation energies, generally producefragment mass distributions peaked at symmetry,average total kinetic energies peaked at syrn-metry and slowly decreasing with asymmetry,and other observed quantities, all of whose gener-al properties are consistent with results of theliquid-drop-model calculations. "" (Measure-ments at low excitation energy do not exist. )

We have chosen the nucleus ' Po to study indetail, as an example of nuclei lighter than radi-um, to determine whether the potential surfacewould exhibit an over-all liquid-drop character.A recent calculation of Bolsterli et a/. ' has indi-cated that the saddle point might occur at an asym-metric shape; if this were confirmed (as has, infact, now occurred in our calculations, see below),the potential energy surface should show signifi-cant deviations from liquid-drop behavior.

The results of our potential energy calculationsfor '"Po are shown in Fig. 12. The figure showsV(A., D), the potential having been minimized withrespect to the other two shape variables o. and o,as for "'U and all other nuclei. The ground state(zero energy) is at D:-7.25 fm; the first barrieris reflection symmetric and occurs at D:-6.6 fm,with V=16 MeV. The second minimum occurs atD:-6.0 fm, with V=11.5 MeV. Again, as in the

10

210po

rA

X=1.5

x=&.0

E

(A

O

CC

5/~

( I,'D CI', 8 A

-10

I I

0 5LENGTH (fm)

I

)0I

f5

PIG. 13. The shape of the ~ Ponucleus at various pointsalong the minimum energy path. The shapes labeled A,8, C, and E, are reflection symmetric P.=1) and corre-spond to neck radius 7.25 fm (ground state), 6.6 frn (firstsaddle), 6.0 fm (second minimum), and 2.0 fm (nearscission), respectively. The shape labeled D is theasymmetric saddle shape and corresponds to the neckradius of 4.5 frn and volume ratio of 1.5.

case of '"U, the second minimum is soft towardsasymmetry and leads to a second saddle at anasymmetry of X-130/80. In this case the secondsaddle consists of a wide region from D -5.0 toD -3.5 in which the energy varies little (only -1.8MeV) and seems to show a slight dip at D -4.5 fm.Beyond this saddle region the energetically fa-vored path leads, via a steep valley, to symmetricmass division at scission.

The second saddle for this case is much higherthan that for "'U and agrees well with the experi-mental barrier height. " We note that the asym-metric saddle is favored by 2 MeV with respectto the symmetric barrier. The second minimumis quite narrow and lies at an excitation energywhich is high relative to the ground state; an inter-esting question is whether this minimum can sup-port an excited state whose wave function is con-centrated in this region.

It is clear that the potential surface V(A, , D)shown in Fig. 12 contains prominent structuresresulting from shell corrections and deviates ap-preciably from the smooth liquid-drop-like po-tential. Still, looking at the region of the surfacefor D = 3.0 fm, we see a resemblance to thesmooth liquid-drop potential; we may understandon this basis, perhaps, why the liquid-drop-


110100

12090

NH

/VL

13080

14070

model calculations initially appeared to be sosuccessful.

In Fig. 13 we show the shapes of the ',O~Po nu-cleus at several points along the minimum pot-tential energy path to scission. %e note especial-ly the asymmetric saddle shape, curve labeled D,and the quite elongated symmetric shape nearscission, reflecting the fact that the nascent frag-ments (",,'Mo at symmetry) are soft, mid-shellnuclei.

A plot of V(X) for several values of D is shownin Fig. 14. The curve labeled D =6.0 corres-sponds to the second minimum, that labeled D= 5.0 corresponds to the onset of the second sad-dle. For D ~ 4.0 symmetric shapes are clearlypreferred. This, as explained in the section on"'U, is the kind of display we use in those caseswhere the entire V(A, D) surface has not been cal-culated; this figure provides a direct compari-son of ' Po with those cases.

Returning briefly to the calculated asymmetric

saddle in "'Po, we offer the following qualitativeexplanation for this phenomenon. The minimumin V(X) at the saddle point, where D = 4.5- 4.0 fm,occurs at X=1.7=132/78 (see Fig. 14). This mini-mum and the shoulder which persists at this valueof A. for cuts taken closer to scission, at D =3.5,3.0, and 2.0 fm, is probably due to the influenceof the double-magic ",',Sn„configuration reachinginto the potential energy surface. Its strength isnot enough to overcome the very strong liquid-drop-model preference for symmetry near scis-sion, but is sufficient to cause the asymmetricminimum in the otherwise flat saddle region.

202E. 82Pb120

This nucleus is another example of nuclei lighterthan radium, for which the fragment mass distri-butions are peaked at symmetry. %e have exam-ined a few slices V(A) at fixed D values, to ascer-tain whether the same general features occur inthe V(X, D) surface for this nucleus as for "'Po.The results are shown in Fig. 15.

The curve labeled D =4.37 shows V(A. ) in theregion of the saddle point, located on the basis

24210po

28

11092

12082

13072

14062

22 26

20

22

&- 20

LLJ

LLJ

18

0= 3,0

12 —t'rD= 6.0

16

101.0 1.2 14 16 1 8

X, VOLUME RATIO

2.0 2.212

1.0 1.4 1.6 1.8 2.0 2.2),VOLUME RATIO

2.4

FIG. 14. The potential energy for 2~0Po as a functionof A, , the volume ratio of the portions of the nucleus oneither side of the neck plane, for several values of D,the radius of the neck in fermis. The energy has beenminimized with respect to u and o and normalized tozero for the ground-state energy.

F'LG. 15. The potential energy for Pb as a functionof A, , the volume ratio of the portions of the nucleus oneither side of the neck plane, for several values of D,the radius of the neck in fermis. The energy has beenminimized with respect to ~ and 0 and normalized tozero for the ground-state energy.

MUSTAFA, NOSE L, AND SC HMITT

of earlier calculations for reflection-symmetricshapes. ' We see that, as in '"Po, an asymmetricshape is preferred. The descent towards scission,i.e., V(A) at smaller values of D, indicates againa steep valley whose minimum is at symmetry.

Thus the results for 2Pb seem to be quali-tatively similar to those for '"Po. In both casesthe potential energy calculations indicate a massdistribution peaked at symmetry, in agreementwith experimental observations for light nuclei.

IV. DISCUSSION AND SUMMARY

The comparison of our results with experiment,as discussed above, shows clearly that the ob-served mass distribution is correlated with thepotential energy surface in the neighborhood ofscission. In this region of the potential surface,fragment structure effects are identifiable andare strongly felt, as we have also shown. We,therefore, conclude that the observed mass dis-tributions in low-energy nuclear fission are strong-ly influenced by specific fragment shell structures.

It is a remarkable coincidence that in '"U, andpossibly also in other actinides, the asymmetryremains almost constant from the saddle point toscission, whereas in "'Po the preferred asym-metry changes drastically between saddle andscission. This comparison appears to us to illus-trate the interesting interplay between the shellsof the fissioning nucleus and those of the nascentfragments in the structure of the potential energysurfaces. The details of this interplay and thespecific manner in which it occurs in the potentialsurface depends to some extent on the form of thesingle-particle potential and on the increase inheight of the intermediate peak in V(z) at the neckplane as the neck radius decreases toward scis-sion. The analytical form we have used (see Sec.I&) appears to be physically reasonable, but thisaspect should ultimately be studied by means ofself-consistent calculations.

Because of the close correlation between ourtheoretical results and experimental observations,it appears as though a fissioning nucleus may "feelout" its potential energy surface, following ratherclosely a minimum potential energy path towardscission. Alternatively, the motion may be morerapid, and the mass parameter may be smoothlyand slowly varying as a function of the shape vari-ables, so that the structure of the potential surfacestill essentially determines the fission path. Theearly formation of the fragment shells is expectedto lead to a collective mass parameter for theseparation degree of freedom which approachesthe reduced mass of the fragments soon after thesecond saddle; thus dynamic effects should have

only little influence on our conclusions drawn fromstatic saddle and scission point properties.

Although the dynamic calculations and solutionof the complete Hamiltonian remain essential, itappears at present that the potential surface playsthe dominant role. This provisional conclusionis based on the discussion of possible dynamiceffects (see above), on the correlations betweenobserved and calculated preferred mass ratiosas discussed above, and on a number of previousworks in which fragment deformations and totalkinetic energies in low-excitation fission havebeen systematized on the basis of simple static-scission models. "

Many experiments and correlations of data overan extended period of time have pointed out theimportance of fragment structure effects in fissionobservables. In our earlier calculations, ' andagain here, we have shown that fragment structureeffects can be identified in the potential surface,and we have indicated how these develop. In theresults reported here, quantitative agreement isfound between the potential energy predictionsnear scission and the observed most probablemass divisions for a variety of nuclei.

ACKNOWLEDGMENTS

The authors gratefully acknowledge helpful andstimulating discussions with Professor W. Greiner.They are pleased to acknowledge the assistanceof M. A. Stephens in carrying out the calculations.

APPENDIX A

12 2

CO+ &Z& = 40&2Z2 9

z 19 2

(A2)

o 2 $ 2 o 2 1p1, +p QyZ~ 4 p2 +2 Q2Z2 ~ (AS)

These four conditions together with the volumeconservation constraint leave us with five inde-pendent potential parameters. Another constraintfollows from the requirement that the functionv„(z) should be practically constant over the wholevolume of the fragment nuclei when the nuclear

The parameters $; and n; appearing in the single-particle potential can be determined with the helpof the following conditions: (1) The nuclear shape[ p(z)] and its derivative [p'(z)] must be continuousat z =O. The nuclear neck should be located atz=0, implying p'(z=0) =0; (2) the single-particlepotential V( p, z) and its final derivatives shouldbe continuous at z =0. Conditions (1) and (2) leadto the following constraints on the potential pa-rameters:


u)„'(z =0) =2((up, '+tdp, ').This leads to:

(A4)

2 (~Pi ~P2 ) ~

o o

1

1 o 2 o 2

82

(A5)

This additional condition then leaves us with fourindependent potential parameters which can beconverted into shape parameters by using Eq. (4);

configuration consists of two well separated frag-ments. In other words, the model should be ableto describe two separated Nilsson models. Thiscondition of a very slowly varying ~„(z) over thefragment volume implies that n; should decreasewith increasing z, . In order to determine this de-pendence we require:

then

(doc]= R,40 8]

oa] —o(u ps

(A6)

Instead of these, however, we choose as shapeparameters, for presentation and discussion ofresults, the physically more meaningful quantitiesD -=radius of the neck in fermis and A. -=volume ra-tio of the nuclear parts on either side of the neckplane. Note A. = j„,pdpdz/f„, pdpdz, where theintegrations are to be extended over indicatedparts of the nuclear volume. (X= 1 correspondsto symmetry. ) The remaining two shape param-eters in this framework are then:

*On leave from the Atomic Energy Center, Dacca,Bangladesh.

)Operated by Union Carbide Corporation for the U. S.Atomic Energy Commission.

f.present address: Institut fur Theoretisehe Physik,Universitat Giessen, Giessen, Germany.

5Work partly supported by the U. S. Atomic EnergyCommission.

~V. M. Strutinsky, Nucl. Phys. A95, 420 (1967); A122,1 (1968); W. D. Myers and W. J. Swiatecki, Arkiv Fysik36, 343 (1967).

S. G. ¹lsson, C. F. Tsang, A. Sobiezewski, Z. Szy-manski, S. Wycech, C. Gustafson, I. L. Lamm, P. Moi-ler and B. Nilsson, Nucl. Phys. A131, 1 (1969); M. Bol-sterli, E. O. Fiset, J.R. Nix, and J. L. Norton, Phys.Rev. C 5, 1050 (1972); H. C. Pauli and T. Ledergerber,Nucl. Phys. A175, 545 (1971); V. M. Strutinsky and H. C.Pauli, in Proceedings of the Second International AtomicEnergy Symposium on Physics and Chemistry of Fission,Vienna, Austria, 1969 (International Atomic EnergyAgency, Vienna, Austria, 1969), p. 155.

3U. Mosel and H. W. Schmitt, Nucl. Phys. A165, 73(1971); Phys. Rev. C 4, 2185 (1971).

4P. Moiler and S. G. NQsson, Phys. Letters 31B, 283(1970); V. V. Pashkevich, Nucl. Phys. A169, 275 (1971).

5P. Holzer, U. Mosel, and W. Greiner, Nucl. Phys.A138, 241 (196S); D. Scharnweber, W. Greiner, andU. Mosel, Nucl. Phys. A164, 257 (1S71).

6A short note on these calculations has previously beenpublished by M. G. Mustafa, U. Mosel, and H. W.Schmitt, Phys. Rev. Letters 28, 1536 (1972).

'C. Y. Wong, Phys. Letters 30B, 61 (1969); G. D.Adeev, P. A. Cherdantsev, and I. A. Gamalya, Phys.Letters 35B, 125 (1S71).

8U. Mosel, J. Maruhn, and W. Greiner, Phys. Letters35B, 125 (1971).

A. similar generalization, with a different shapeparametrization, is presently being developed byJ. Maruhn and W. Greiner, Z. Physik 251, 431 (1972).

D. A. Arseniev, A. Sobiezewski, and V. G. Soloviev,Nucl. Phys. A139, 269 (1969).~1C. Gustafson, I. L. Lamm, B. Nilsson, and S. G. Nils-

son, Arkiv Fysik 36, 613 (1S67).P. A. Seeger, CERN Report No. CERN 70-30, 1970

(unpublished) .~3T. Johansson, S. G. ¹ilsson, and Z. Szymanski, Ann.

Phys. (Paris) 5, 377 (1970).~4U. Mosel, phys. Rev. C 6, 971 (1972).5This is not the case in the calculations of Bolsterli

etal. (Ref. 2) where only a one-dimensional cut into theasymmetric degree of freedom at the symmetric saddlehas been calculated. At least a two-dimensional calcu-lation is necessary tc determine the saddle point.

~If the ground state of 6U has a hexadecapole deform-ation, as indicated in recent Coulomb excitation experi-ments at this Laboratory [F. K. McGowan, C. E.Bemis, Jr. , J. L. C. Ford, Jr. , W. T. Milner, R. L.Robinson, and P. H. Stelson, Phys. Rev. Letters 27,1741 (1971)], then the location of the ground state would

be moved to a higher value of D; the surface around and

beyond the second minimum would not be affected.~7H. W. Schmitt, J. H. Neiler, and F. J. Walter, Phys.

Rev. 141, 1146 (1966), and references therein.N. L. Lark, G. Sletten, J. Pedersen, and S. Bjgfrnholm,

Nucl. Phys. A139, 481 (1969).19R. Brandt, S. G. Thompson, R. C. Gatti, and

L. Phillips, Phys. Rev. 131, 2617 (1963).2 K. F. Flynn, E. P. Horwitz, C. A. A. Bloomquist,

R. F. Barnes, R. K. Sjoblom, P. R. Fields, and L. E.Glendenin, Phys. Rev. C 5, 1725 (1972).

21W. John, E. K. Hulet, R. W. Lougheed, and J. J.Wesolowski, Phys. Rev. Letters 27, 45 (1971).

J. P. Balagna, G. P. Ford, D. C. Hoffman, and J. D.Knight, Phys. Rev. Letters 26, 145 (1971).

J, R. Nix, W. J. Swiateeki, Nucl. Phys. 71, 1 (1965);J. R. Nix, Nuel. Phys. A130, 241 (1969).

24F. Plasil, D. S. Burnett, H. C. Britt, and S. G.Thompson, Phys. Rev. 142, 696 (1966).

MUSTAFA, MQSEL, AND SCHMITT

5L. G. Moretto, S. G. Thompson, J. Routti, and R. C.Gatti, Phys. Letters 388, 471 (1972).

268ee, e.g. , H. W. Schmitt, in Proceedings of the Sec-ond International Atomic Energy Symposium on Physics

and Chemistry of Fission, Vienna, Austria, 1969 (seeRef. 2), p. 67, and references therein; also H. W.Schmitt and U. Mosel, Nucl. Phys. A186, 1 (1972).

PHYSICAL REVIE W C VOLUME 7, NUMBER 4 A PRIL 1973

92Neutron Resonance Parameters of Mo~

Q. A. WassonBrookhaven National Laboratory, Upton, Nese York 11973,


B.J. AllenAustralian Atomic Energy Commission, Lucas Heights, Australia,and Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

R. R. WintersDenison University, t"ranville, Ohio 43023,


R. I. Macklin and J. A. HarveyOak Ridge National Laboratory, Oak Ridge, Tennessee 37830

(Received 21 August 1972)

Neutron transmission and total neutron-capture yields were measured using the time-of-flight technique at the Oak Ridge electron linear accelerator facility using enriched samplesof ~2Mo. A total of 42 resonances were observed for neutron energies less than 32 keV. 12are assigned to s-wave interactions, while 23 are assigned to p-wave interactions. Theresonant energies, neutron widths, radiation widths, and spins are deduced. The averages- and p-wave radiation widths are equal to 0.178+ 0.015 eV and 0.24+ 0.03 eV, respectively.The s-wave neutron strength function is (0.65+ 0.26) && 10 4, while the p-wave strength func-tion is (3.3+1.1)x 10 4.

I. INTRODUCTION

A recent development in the field of radiative neu-tron capture is the observation of the importanceof single-particle or valency contribution to they-ray decay of the neutron resonances for nucleinear the peak of the 3p neutron strength function. ' 4

In a separate paper' capture y-ray spectral mea-surements are reported for resonant neutron cap-ture in the target nucleus 'Mo for neutron energiesless than 25 keV. Qne purpose of the present paperis to supply the neutron resonance parameters re-quired to calculate the valency model contributionsto the y-ray transition probabilities for this nu-cleus. Previous measurements' of the total neu-tron cross section, which employed samples ofnatural isotopic composition, identified only a few

resonances in "Mo. Total neutron-capture mea-surements by Weigman, Rohr, and Winter' usingenriched samples were limited to neutron energiesless than 10 keg.

The present work reports the results of neutrontransmission and total neutron-capture experi-ments performed at the Qak Ridge electron linearaccelerator (ORELA). Neutron resonance parame-ters for the target nucleus "Mo are deduced forneutron energies less than 32 keV where only s-and p-wave neutron interactions are significant.The required parameters are the neutron widthI'„, the total radiation width I'„, and the spin andparity of the resonance J~.

The parities of the resonances are determinedfrom the shapes of the transmission dips. Thes-wave resonances are identified by the asymmet-

Documents

Asymmetry in Nuclear Fission