to fission or not to fission

7/25/2019 To Fission or Not to Fission

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arXiv:1607

.00353v1

[nucl-th]

1Jul2016

To fission or not to fission

K. Pomorski and B. Nerlo-Pomorska

Maria Curie Sklodowska University, Lublin, Poland

F.A. Ivanyuk

Institute for Nuclear Research, Kiev, Ukraine(Dated: July 4, 2016)

The fission-fragments mass-yield of 236U is obtained by an approximate solution of the eigen-value problem of the collective Hamiltonian that describes the dynamics of the fission processwhose degrees of freedom are: the fission (elongation), the neck and the mass-asymmetry mode.The macroscopic-microscopic method is used to evaluate the potential energy surface. The macro-scopic energy part is calculated using the liquid drop model and the microscopic corrections areobtained using the Woods-Saxon single-particle levels. The four dimensional modified Cassini ovalsshape parametrization is used to describe the shape of the fissioning nucleus. The mass tensor istaken within the cranking-type approximation. The final fragment mass distribution is obtained byweighting the adiabatic density distribution in the collective space with the neck-dependent fissionprobability. The neck degree of freedom is found to play a significant role in determining that finalfragment mass distribution.

PACS numbers: 21.10.Dr, 24.10.Cn, 25.85.-w

Keywords: nuclear fission, fission fragment mass distribution

I. INTRODUCTION

A very stringent test of any theoretical model whichdescribes the nuclear fission process should be a properreproduction of the fission fragments mass distribution.The goal of the present paper is to obtain such a dis-tribution by an approximate solution of the eigenvalueproblem of the 3-dimensional collective Hamiltonian withdegrees of freedom corresponding to elongation, neck for-mation and mass asymmetry of the nuclear shape. Thepresent model is similar to the 2-dimensional one of Refs.[1,2] but the non-adiabatic and dissipative effects are not

taken here into account since their effect is rather smallfor low-temperature fission. The potential energy sur-face (PES) is obtained in the present work using themacroscopic-microscopic method with the liquid dropmodel for the macroscopic part of the energy while themicroscopic shell and pairing corrections are calculatedusing the Woods-Saxon (WS) single-particle levels [3].The shape of the fissioning nucleus is described by thefour dimensional modified Cassini ovals (MCO)[3,4]. Itwas shown in Ref. [5] that the MCO describe very wellthe so-called optimal nuclear shapes obtained through avariational description [6], even those close to the scis-sion configuration. The mass tensor is taken within the

cranking-type approximation (confer e.g. Sec. 5.1.1 ofRef.[7]). The Born-Oppenheimer approximation (BOA)is used to describe the coupling of the fission mode withthe neck and mass asymmetry degrees of freedom. It willbe shown that, in order to obtain a fission-fragment mass

Electronic address: [email protected] address: [email protected] address: [email protected]

distribution in agreement with the experimental data,that fission probability should depend on the neck size.

The paper is organized in the following way. First weshortly present the details of our theoretical model, thenwe show the collective potential energy surface evaluatedin the macroscopic-microscopic model for 236U and thecomponents of the mass tensor. The calculated fissionfragments mass distribution is compared with the experi-mental data in the next section. Conclusions and possibleextensions and applications of our model are presentedin Summary.

II. COLLECTIVE HAMILTONIAN

A. Shape parameterization

We define the shape of fissioning nucleus by the pa-rameterization developed in[3]. In this parameterizationsome cylindrical co-ordinates{, z} are related to thelemniscate co-ordinates system{R, x} by the equations

= 1

2

p(x) R2(2x2 1) s ,

z =

sign(x)

2 p(x) +R2(2x2 1) +s,p2(x) R4 + 2sR2(2x2 1) +s2,

0 R ,1 x 1. (1)The co-ordinate surfaces of the lemniscate systemR(x) =R0 are the Cassini ovals (see the bottom of Fig. 1) withs R20, where s is the squared distance between thefocus of Cassinian ovals and the origin of coordinates.

The deviation of the nuclear surface from Cassini ovalsis defined by expansion ofR(x) in series in Legendre poly-
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2

0.5

0.25

0

0 0.5

1

0.95 1.0

FIG. 1: Examples of nuclear shapes in, 1, 4 parametriza-tion. The solid lines correspond to 4=0 while the dashedand dotted curves to 4=0.2 and4=-0.2 respectively.

nomials Pn(x),

R(x) = R0[1 +n

nPn(x)] , (2)

where R0 is the radius of the spherical nucleus. Thecylindrical co-ordinates {, z}are related to{, z} by

/c, z (z zcm)/c , (3)

where zcm is the z-coordinate of the center of mass ofCassini ovaloid (2) and the constant c is introduced inorder to insure the volume conservation.

Instead of, it turns out convenient to introduce an-other parameter, , which is defined so, that at = 1the neck radius turns into zero for any value of all otherdeformation parametersn,

=

1

4 [(1 +n n)2

+ (1 +n (1)n

n)

2

]

++ 1

2 [1 +

n

(1)n2n(2n 1)!!/(2nn!)]2. (4)

The parameters and n are considered as the defor-mation parameters. Examples of the shapes in Cassiniparameterization are shown in Fig. 1.

B. Born-Oppenheimer approximation

Below we use the following collective coordinates to

describe the fission dynamics:

q1 =R12 , q2 = a=

V1 V2V1+ V2 , and q

3 =4. (5)

Here R12 is the distance between the mass center of thenascent fragments in units of the radius R0 of the corre-sponding spherical nucleus, while ais the mass asymme-try coordinate. V1 andV2 are the volumes of the frag-ments and4 describes the neck degree of freedom whenR12 is kept constant (see Ref.[4]).

With these coordinates the classical energy of the systembecomes

Hcl=1

2

i,j

Mijqiqj +V({qi}) , (6)

whereMij andV({qi}) denote the inertia tensor and thepotential energy, respectively.

The quantized form of this Hamiltonian is the following:

H= 22

i,j

|M|1/2 qi

|M|1/2Mij qj

+V({qi}),

(7)where|M| = det(Mij) and MijMjk =ki.

The eigenproblem of this Hamiltonian will be solvedhere in the Born-Oppenheimer approximation in whichone assumes that the motion towards fission is muchslower than the one in the two other collective coordi-nates. In such an approximation the Hamiltonian couldbe written as follows:

H(R12

, a, 4

)

Tfis

(R12

) + Hperp

(a, 4

; R12

) . (8)

Here Tfis is the fission mode kinetic energy operator

Tfis(R12) = 2

2

R12M

1(R12)

R12, (9)

whereM(R12) is the average inertia related to the fission

mode and Hperp is the collective Hamiltonian related tothe neck and mass asymmetry coordinates. The eigen-function of Hamiltonian (8) can be written as a product:

nE(R12, a, 4) = unE(R12)n(a, 4; R12), (10)

wheren are the eigenfunctions ofHperp:

Hperp n(a, 4; R12) = en(R12)n(a, 4; R12) , (11)

and they are evaluated for each mesh-point value in theR12direction. Using the above relations one can rewritethe eigenequation of the Hamiltonian(8) in the followingform:

Tfis+en(R12)

unE(R12) =E unE(R12) . (12)

The approximate solution of the above eigenvalue prob-lem can be obtained using the WKB formalism. The en-ergiesen(R12) in Eq. (12) define the fission potential fordifferent channels, what is important when one describesthe nonadiabatic fission process in the coupled channelsapproach [2]. In the following we shall take only the low-est energy channel, what corresponds to the adiabatic ap-proximation. Within this approximation the wave func-tion of the fissioning nucleus is written in the form of aproduct of the wave functionu0E(R12) describing the mo-tion towards fission and the function0(a, 4; R12) whichcorresponds to the lowest eigenenergye0of the Hamilto-nian (11). The probability of finding of a nucleus, for a


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3

given value ofR12, in the given (a, 4) point is equal to|0(a, 4; R12)|2.

In Refs. [1, 2] is shown how to go beyond the BOAand include nonadiabatic and dissipative effects, but inthe following we are going to omit such effects, which areexpected to be small at low temperatures and limit ourdiscussion to the effect of the neck degree of freedom onthe fission-fragment mass distribution.

III. NUMERICAL RESULTS

The potential energy surface is evaluated for 236Uat zero temperature within the macroscopic-microscopicmodel in which the macroscopic part of the energy isobtained using the liquid drop formula and the micro-scopic shell and pairing corrections are calculated usingthe Woods-Saxon single-particle potential. All param-eters of the calculation are described in Ref. [4]. Thepotential energy was calculated in 4-dimensional spaceof deformation parameters , 1, 4, 6 and minimized

then with respect to 6.The mass parameter Mij(q) for the fission process iscommonly calculated by the Inglis formula

Mij(q) = 22m

0|/qi|mm|/qj|0Em E0 , (13)

where |0 and |m denote the ground and an excited stateof the system.

In the case that the ground and the excited states ofthe system are described by the BCS approximation, theMij(q) is given by [9],

Mij = 22

|H/qi

|

|H/qj

|

(E+E)3 2

+ Pij,

(14)where the term Pij stands for the contribution due tothe change of occupation numbers, when the deformationvaries. TheE, u and v in (14) are the quasi-particleenergies and coefficients of the Bogolyubov-Valatin trans-formation correspondingly, and uv+ uv.

Unfortunately, the expression (14) has the very un-pleasant feature that it does not turn into the mass pa-rameter of a system of independent particles when thepairing vanishes, 0. More precisely, the non diago-nal sum over single-particle states in (14) does turn intothe mass parameters of the system of independent par-ticles when 0, but, even worse, the diagonal sumgoes to infinity in that limit (it is proportional 1/2, asdemonstrated in [9]). Thus, at some points in the defor-mation space, where the density of single-particle states isvery low, the mass parameter (14) becomes unreasonablylarge. The same happens in excited systems, when thetemperature is close to its critical value Tcrit at whichthe pairing gap disappears.

One should also keep in mind that the diagonal contri-bution to the sum in (14) comes from the matrix elements

between the ground state and the pair excited states thatcorrespond to the particle number, different from that ofthe ground state. In a particle number conserving theorysuch contribution could not appear.

In order to avoid the problems related to the diagonalcontribution to (14) we have omitted in (14) the diago-nal matrix elements |H/qi| and taken into accountonly the non-diagonal part of (14),

Mij = 22

=

|H/qi||H/qj |(E+E)3

2.

(15)The inertia tensor (15) is evaluated in the 3-dimensionalspace of deformation parameters , 1, 4. For each valueof 4 and the potential energy and the components ofmass tensor were transformed from , 1 to R12, a coor-dinates defined in Eq. (5). The potential energy surface(PES) related to the spherical liquid drop energy (upperrow) and the MR12R12 component of the inertia tensor(middle row) as well as the neck radius = rneck/R0(lower row) are plotted in Fig. 2 on the (Af, 4) plane

for three different values of the relative distances of thecenter of the fragmentsR12. HereAfis the atomic massof fission fragment.

In the following we would like to describe the wayin which one has to obtain the fission fragment massyield after solving the quantum mechanical problem ofthe collective Hamiltonian which describes the fissionprocess in the three dimensional space (3D) composedof the following deformation parameters:

R12 distance between the mass center of the fissionfragments,

a = [Af(1) Af(2)]/[Af(1) +Af(2)] the mass asym-metry parameter, where Af(1) and Af(2) are the

mass numbers of the fission fragments,4 hexadecapole correction to the Cassini ovals [3].

First one has to prepare a set of the 2D mass distribu-tions

|(a, 4; R12)|2 = |0(a, 4; R12)|2 (16)on the plane (a, 4) by solving eigenproblem of a corre-sponding 2D collective Hamiltonian for fixed elongationsR12[8]. One has to bear in mind that it is very unlikely,that fission occurs at some fixed R12or when the systemreaches the scission line/surface. The problem is muchmore complicated and one has to involve into considera-

tion the size of the neck.Looking at the integrated over4probability distribu-tions for 236U presented in the top part of Fig. 3

w(a, R12) =

|(a, 4; R12)|2d4 , (17)

one can not see any qualitative change with respect to theresults which we have published in Ref. [2] for the cal-culation made in the (R12, a) plane. Both distributions,i.e. the one corresponding to Af(1) = 140 for smaller


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4

FIG. 2: (Color online) Macroscopic-microscopic PES (upper row) and the relative distance componentMR12R12 of the inertiatensor (middle row) and the neck radius = rneck/R0 (lower row) on the (Af, 4) plane for different values of the elongationR12.

R12and the one for Af(1) = 132 atR12close to the scis-sion line are only slightly broader. So, the problem toreproduce the experimental distribution of fission frag-ments, seen in the 2D space [2], remains also in the 3Dspace. The only solution is to assume that fission occurswith a certain probability before (or after) reaching thecritical elongation Rcrit12 . Depending on the neck radiusa fissioning nucleus has to make its choice to fission ornot to fission. When itdecides for fissionit would leavethe phase-space of collective coordinates. Of course thisis not a Hamlet dilemma, where there is only the choicebetweenyesor no. We are rather faced here with a statis-tical problem and the answer yesis given with a certainprobability which one then will have to take into accountin the distribution probability (16) in the phase space.

Following such an assumption a part of the events(read trajectories in the Langevin approach, or distri-butions in our quantum mechanical model) disappearsfrom the phase-space and leads to a kind of weightingof the mass distribution corresponding to the differentelongationsR12. To do this one has to evaluate the neck

radius in the whole 3D space. The neck radius parame-ter = rneck/R0 is plotted in the lower row of Fig.2 onthe (Af, 4) plane for three different values of the elon-gation parameter R12. The slight wiggles in Fig. 2 arecaused by the approximate minimization with respect tothe 6 deformation parameter. It is seen that on av-erage the neck radius decreases with growing R12. Fora constant R12 the neck radius varies strongly with 4and Af. One commonly agrees that fission takes placewhen the neck radius becomes of the order of the size ofa nucleon. This is the case for 0.2, which is real-ized at R12/R0 = 2.0 for 4 =

0.18; R12/R0 = 2.25

for 4 = 0, and R12/R0 > 2.5 for 4 = 0.18 and theasymmetry parameter a0.2. From the optimal shapeapproach [6] one knows that the scission shape corre-sponds to rneck 0.3R0 and Rcrit12 2.3R0. In the caseof the Cassini parametrization used in the present papertherneck can be somewhat smaller.

One could try to parametrize the neck-rupture proba-


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5

120 130 140 150 160

0

2

4

6

8

10

12 1.8

1.9

R12/R

0=2.0

2.1

2.2

2.3

w(

a.u.)

236U,Pneck=1

120 130 140 150 160

0

2

4

6

8

10

12 1.8

1.9

R12/R

0=2.0

2.1

2.2

2.3

w(

a.u.)

Fragment mass number

236U

FIG. 3: (Color online) Probability distribution (17)(top) and

the fission probability (21) (bottom) calculated at few fixedelongations of the fissioning nucleus.

bilityPin the following form:

P(a, 4, R12) = k0

kPneck() , (18)

wherek is the momentum in the direction towards fission(or simply the velocity along the elongation coordinateR12), while = (a, 4, R12) is the deformation depen-dent relative neck size. The scaling parameterk0, playsno essential role, and will disappear from the final expres-

sion of the mass distribution when one will normalize it.The geometry dependent part of the neck breaking prob-ability is taken in the form of a Fermi function:

Pneck() =

1 +e0

d

1. (19)

The parameters0 andd have to be fixed by comparingthe theoretical fission fragment mass distributions withthe experimental ones. Our goal is to fix these parame-ters in a kind of universal way, independent of the spe-cific fission reaction that one wants to investigate. Thepresent investigation has to be treated only as a first at-tempt in this direction.

The momentum k which appears in the denominatorof Eq. (18) has to ensure that the probability dependson time in which one crosses the subsequent intervals inR12 coordinates: t = R12/v(R12), where v(R12) =k/M(R12) is the velocity towards fission. The value ofk depends on the differenceE V(R12) and on the partof the collective energy which is converted into heatQ:

2k2

2M(R12)=Ekin= E Q V(R12). (20)

In the quantum mechanical picture the heat Q can bereplaced by the imaginary part of the collective potential[2]. In the Langevin picture the method should be almostthe same but one has also to work at least in the 3Dspace. In our present calculations we have put Q = 0,i.e., we assumed a complete acceleration scenario: nodissipation takes place, which is reasonable since at lowexcitation energies the friction force is very week.

The M in (20) is the cranking inertia relative to theR12 deformation parameter. In principle, we used thedefinition of cranking inertia, but for the reasons ex-plained above the contributions from the diagonal matrixelements were removed.

The fission probability w at a given R12 and a will begiven by the integral:

w(a, R12) =

4

|(a, 4; R12)|2P(a, 4, R12) d4 . (21)

The dependence of the fission probability (21) on themass asymmetry is shown in the bottom part of Fig. 3fora few values ofR12. One observes that due to the Fermifunction in (19) the contribution of larger R12 (smallernecks) is enhanced and contributions from smaller R12are suppressed.

From the maps of the potential energy surface shownin the top part of Fig. 2 one observes that for R12 2.1the neck parameter is4= 0.05 while forR12 2.2 theminimum at the PES is at4= 0.15. This means thata large part of the distribution probability |(a, 4; R12|2will undergo fission also at smaller elongations and onehas to subtract this part from the phase-space, i.e. todiminish the initial distribution by subtracting the eventswhich have already fissioned. The final (measured) massdistribution of the fission fragments will be the sum of

those subtracted events.Such an approach means that the fission process should

be spread over some region ofR12and that for given R12at fixed mass asymmetry one has to take into account theprobability to fission at previousR12points. I.e., one hasto replacew(a, R12) by

w(a, R12) = w(a, R12)

1

R12R12

w(a, R12) dR12

w(a, R12) dR

12

.(22)

The effect of the replacement (22) is demonstrated inFig.4. It is seen there that this replacement substantially

reduces the magnitude of the fission probability at largeR12.

The mass yield will be the sum of all partial yields atdifferentR12:

Y(a) =

w (a, R12) dR12

w(a, R12) dR12da. (23)

As it is seen from(23) the scaling factork0in the expres-sion forP, Eq. (18), has vanished and does not appear in


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6

1.7 1.8 1.9 2.0 2.1 2.2 2.3

0

2

4

6

8 0.05

asymm.=0.10

0.15

w,w'

(a.u.)

R12/R

0

236U

FIG. 4: (Color online) Comparison of the fission probabilities(17) (dashed line) and (22) (solid line) for a few values of themass asymmetry a=asymm.

the definition of the mass yield. Our model will thus onlyhave two adjustable parameters, 0 and d, that appearin the neck-breaking probability (19).

80 100 120 140 1600

2

4

6

8

Yield(%)

Fragment mass number

235U+n

th

FIG. 5: Comparison of the measured mass distribution offission fragments (points) in the reaction 235U +nth [10] with

the values (23), calculated with 0

= 0.16, d = 0.09 (solidline).

A comparison of the measured [10] and here calculatedfission fragment mass distributions is shown in Fig. 5 forthe thermal neutron induced fission of 235U.

One can see that the calculated mass distribution isvery close to the experimental values. The double-peakstructure, the position and the relative magnitude of the

peaks are reproduced rather well.

IV. CONCLUSIONS

The extended Cassini ovals deformation parametersand the macroscopic-microscopic model (ELD plus the

WS single-particle potential) yields for 236U a PES withan asymmetric fission valley corresponding to Af 140when the relative distance between the fragment masscenters is smaller than R12 = 2.3R0. At larger elonga-tions one observes a sudden jump of the maximum of thedistribution toAf 132 what causes severe problems ina correct reproduction of the data.

We have shown that the three-dimensional quantummechanical model which couples the fission, neck andmass asymmetry modes is able to describe the mainfeatures of the fragment mass distribution when theneck dependent fission probability is taken into account.The obtained mass distribution is slightly shifted, by

approximately 2 mass units, towards symmetric fissionas compared with the experimental mass yield, butreproduces nicely the structure of the distributionobserved in the experiment. This shift could be partlydue to a too large stiffness of the LD energy in the massasymmetry degrees of freedom and/or to a lack of thenonadiabatic effects (beyond the Born-Oppenheimerapp.) [2] which makes the distributions slightly widerthan the sole adiabatic ones. Also the energy dissipation,not taken into account in the present investigation, couldmodify somewhat the theoretical distribution.

Acknowledgments

The authors are very grateful to Drs. ChristelleSchmitt and Johann Bartel for the careful reading of themanuscript and the valuables comments. One of us (F.I.) would like to express his gratitude to the Theoret-ical Physics Division of UMCS for the hospitality dur-ing his stay at Lublin. This work has been partly sup-ported by the Polish National Science Centre, grant No.2013/11/B/ST2/04087.

[1] B. Nerlo-Pomorska, K. Pomorski, E. Werner, Int. Conf.

Fifty Years Research in Nuclear Fission, Contributed Pa-pers, Berlin, 1989, p.63

[2] B. Nerlo-Pomorska, K. Pomorski and F.A. Ivanyuk, ActaPhys. Polon. B Sup. 8, 659 (2015).

[3] V.V. Pashkevich, Nucl. Phys. A 169, 275 (1971).[4] V.V. Pashkevich and A.Ya. Rusanov, Nucl. Phys. A810,

77 (2008).[5] N. Carjan, F.A. Ivanyuk, V.V. Pashkevich, Physics Pro-

cedia 31, 66 (2012 ).[6] F.A. Ivanyuk, K. Pomorski, Phys. Rev. C 79, 054327

(2009).

[7] H.-J. Krappe and K. Pomorski, Theory of Nuclear Fis-

sion, Lecture Notes in Physics 838, 2012, Springer Verlag.[8] B. Nerlo-Pomorska, K. Pomorski, Zeit. Phys. A309, 341

(1983).[9] M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M.

Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44, 320(1972).

[10] S. Zeynalov, V. Furman, F.-J. Hambsch, M. Florec,V.Yu. Konovalov, V.A. Khryachkov and Yu.S. Zamyat-nin, in Proc. the 13th Int. Sem. on Interaction of Neu-trons with Nuclei (ISINN-13), JINR Dubna, 2006, p. 351.

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