transfer-induced fission in inverse kinematics: from...

Transfer-induced fission in inverse kinematics: from fission-fragment to scission-fragment characterization

F. Farget, O. Delaune, X. Derkx, C. Golabek, T. Roger, A. Navin, M. Rejmund, C. Rodriguez-Tajes, C. Schmitt GANIL, France K.-H. Schmidt, B. Jurado CENBG, France D. Doré, M. Delphine de Salsac SPhN, France J. Benlliure, M. Caamaño, E. Casarejos, D. Cortina, B. Fernandez-Dominguez, D. Ramos USC, Spain L. Audouin, C.-O. Bacri, IPNO, France L. Gaudefroy, J. Taieb CEA DIF, France A. Heinz Chalmers U., Sweden

ARTHUR C. WAHL Fission of 235U, 233U, 239Pu, and “‘Cf

0.0 : 115 120 125 130 1x5 140 145 ml l55 160

to ~"","","","","","",,"',"",""J

-10 r,,,,‘,,,,‘,,‘.“,“‘,,‘,‘,,,,‘.,,.’.’,”’,,’l ll5 lzo 125130135140145150155160

A’, 236-A’

FU239T 10

2 x 2

0.5

0.0 ll5 lzo 125130135140145150 155 160

0.5 -

32 oo- 1 $1 -1

-0.5 -

4.0 t,““,““.‘.“,‘.““,,““.‘.,,“.“““‘,1 115 120 125130135!40!45150 155160

A’, 240-A

U233T l.0 c”“,““,““,““,“““““““,““,““,

-to 115 120 125 130 135 140 145 150 155 160

A’, 234-A

cF252s 1.0 ~"",:"',"","","","","","","",""~

0.0 ,,1,...I...,I.,..I....I....I....I....I....l....j l2o125130135140145150155160165m

120125 130 135 140 145 150 I55 160 65 170

A’, 252-A

Figure 4. &model representation of nuclear-charge polarization and dispersion as AZ(A) and RMS(A) vs A’ functions [see Eqs. (2), (S), (8), and (8a)l. 0 , heavy-fission-product, and Cl, light-fission-product experimental A&4) = .&4) - AI(ZF/AF) and RMS(A) [Eq. (8a)] both from Tables I-IV, as in Fig. 3. -, heavy-fission-product, and dots e l l ), light-fission-product AZ(A) and RMS(A) [Eq. (8a)] both from AL-model calculations. ---, AZ(A) and rms(A) functions [Eqs. (5) and (8)] used for Z,,-model calculations, as in Fig. 3.

FN(Z) = F.v for even N

FN(Z) = l/fi,~ for odd N

&ZL) = zL[AF/zFl - m’(zH,),

(ZHC = z, - z,)

&zH) = zH[AF/zF] + aAt

(9d

W)

fL4’(ZH > 50) = C + (S/2){ln[Y(ZH + l)] - ln[ Y(ZH -l)]}. (9e)

AA’ (ZH = 48, 49, 50) are determined independently, or fixed values are assumed.

&.‘f(Z,/2) = 0 @f)

10 Atomic Data end Nuclear Data Tables. Vol. 39, No. 1, May 1999

Fission fragments : complex witnesses of the fission process

J. Phys. G: Nucl. Part. Phys. 35 (2008) 035104 A V Karpov et al

(a) (b)

Figure 2. Macroscopic (a) and macro-microscopic (b) potential energy surface for the 238Unucleus in the coordinates (R, !). The potential energy is obtained for " = 0 and # = 0.35. Themacroscopic part is normalized to zero for the spherical shape of the compound nucleus.

potential) is located at ! = 0 and somewhere in between r = rgs and r = rgs + $r . Thesecond term in (4) is responsible for the shell effects in the nascent fragments. They showup in the region of large deformations, while the compound-nucleus shell effects vanish withdeformation. The fragment shell correction has the amplitude Afr, and it is modeled again bya cosine function with exponential damping. The corresponding shell-oscillation length scaleand damping factor are %

(!)fr and l

(!)fr , respectively.

An example of the potential energy (both the macroscopic and full macro-microscopic)for the 238U nucleus is shown in figure 2. The same potentials obtained within the schematicmodel are shown in figure 3. The typical values of the ratios extracted above from the standardmacro-microscopic model in the uranium region of nuclei are: %

(r)CN

!%̃ ! %

(!)CN

!%̃ ! 0.3 and

%(!)fr

!%̃ ! 0.1. These values were assumed in figure 3. Comparing with the realistic macro-

microscopic potential we see that the schematic model gives reasonable potential energiesdescribing the main features of the realistic ones.

3. Results and discussions

For the determination of the saddle point we applied the ‘flooding’ procedure [9, 14]. Stepby step ‘water’ is added to the ground-state region, and we control the ‘wet’ region of thepotential energy surface. The procedure is finished when the scission region becomes wet.This minimal height of the water column equals the value of the fission-barrier height.

It was pointed out that while the compound nucleus deforms from its ground state toscission, a neck in the nuclear shape appears and the mononucleus decays into two fragmentswhen the system looses stability against neck variation. With increasing deformation, theoscillating compound-nucleus shell correction vanishes, and the shell effects associated withthe nascent fragments establish. Consequently, the dependence of the shell correction onthe elongation r gradually disappears for the highly deformed mononucleus, and it vanishescompletely asymptotically for two well-separated nuclei.

8

A*1,Z*1,V1/A*2,Z*2,V2 Afiss, Zfiss,E*

Karpov et al. JPG 35(2008) A. Wahl et al. ANDT 39 (1988)

Deformation of the fissioning system: Mass asymmetry, influence of shell effects, neutron and proton sharing, dissipated energy

1

CN

ARTHUR C. WAHL Fission of 235U, 233U, 239Pu, and 252Cf

U235T U233T

2.5

I? 2o

l.5

1.0

0.5

0.0

4.0

3.5

3.0

2.5

20

1.5

1.0

0.5

0.0

70 80 90 100 lx) 120 130 140 150 160 I70

A

PU239T cF252s

70 80 90 100 1lO 120 130 140 150 160 l70 70 80 90 100 110 120 130 140 150 160 l70

A A

4.0

3.5

3.0

2.5

20

1.5

1.0

0.5

0.0

2.5

IP 2o

15

Figure 1. Average number of prompt neutrons emitted, VA, to form fission products with mass number A. The derived function used (sea text and Tables I-IV) is shown by the solid line. Symbols represent experimental values of the average number of neutrons emitted by fragments, 51, plotted against Ar- Sr = A, AFbeing the average fragment mass number before prompt-neutron emission. U235T: 0 (Ref. 23), A (Ref. 24), 0 (Ref. 25), V (Ref. 26). U233T: 0 (Ref. 27, A (Ref. 25). PU239T: 0 (Ref. 28), A (Ref. 25). CF252S: O(Ref. 29) A (Ref. 30), X (Ref. 31), 0 (Ref. 32), V (Ref. 33).

rameters are derived for heavy products, complemen- tary values being used for light products, because the heavy mass and charge peaks remain in approximately the same positions for the four fission reactions, as will be discussed. Thus, model parameters for various fission reactions may be sufficiently similar for ready compari- son and for estimation of some parameters for fission reactions with relatively few data.

The dispersion in yields for both the Z, and AZ, models is assumed to be Gaussian, but yields are modu- lated by proton- and neutron-pairing effects. The effects are applied by multiplication or division of Gauss- ian yields by Fz and by pN, the average even-odd-

proton and -neutron factors, respectively’ (previously called2”‘94,37 EOZ and EON). The Gaussian width parameters for the Z,, and A’p models are represented by the symbols uz and aA,, respectively; the values of these parameters are equal to the root-mean-square (rms) values for Gaussian dispersions corrected for grouping.38

u = [(rms)2 - l/12)]“? (4) It is convenient to compare Z, and Aj, values with

those for unchanged-charge division, ZucD = A’(ZF/AF) and &cn = Z(&/Zr), and the differences,

AZ= <zp- &JCD)H = &CD- Z&L (5)

AA'= (A~-AJCD)H = MJ~D-QL, (6)

6 Atomic Data and Nuclear Data Tables, Vol. 39, No. 1, May 1999


Scission: velocities and TKE inform on scission deformation 2


3

Figure 1. The data points mark the measured elemental yields (left column) and average total kinetic energies (right column) as a function of the nuclear charge of the fission fragments for several fissioning nuclei. Only statistical errors are given. They are not shown when they are smaller than the symbols. The total kinetic energies are subject to an additional systematic un-certainty of 2 %, common to all data. The full lines show descriptions with the model of inde-pendent fission channels with the parameters given in table 1. Dashed lines depict the contribu-tions of the individual channels. The sequence from symmetry to largest asymmetry in the yields is: super long, standard I, standard II. The sequence from lowest to highest TKE is: super long, standard II, standard I. (See text for details.)

3


3


C. Boeckstiel et al. NPA 802 (2008)

As further n evaporation is not modifying in average the velocity vector


β

ν

Separated fragments: neutron evaporation, final fragment distribution 3


Wilkins et al. PRC 14 (1976) M. Caamaño et al. PRC 88 (2013)

Fission fragment distribution predictions

•  Fission-fragment distributions are a key observable for benchmarking code predictions

•  Transfer-induced fission in inverse kinematics has recently brought up the experimental characterization in (Z,A) of the fragments

•  The evolution of the fissioning nucleus up to the scission point may be tracked by the measure of

–  TKE – Neutron multiplicity

•  We will see in the following slides how to characterize in (Z,A) the fragments at scission

Transfer-induced fission in inverse kinematics

ZpApP Zt

AtT

Zt 'At 'T '

Zp 'Ap 'P '

Zf 1'Af 1F1

Zf 2Af 2F2

•  10 actinides produced •  E* distribution •  Full resolution in (Z,A) of fragments •  TKE •  Détermination of scission fragments

238U +12C @ 6.1 MeV/u


SPIDER

VAMOS

ΔE-E,θ

Bρ-ToF-ΔE-E

S. Pullanhiotan et al., NIM 593 (2008) 343 M. Rejmund et al., NIMA 646 (2011) 184

EXOGAM


SPIDER C. Rodriguez-Tajes et al., PRC (2014) 024614

Isotopic distribution of fission fragments

M. Caamaño et al., PRC 88 (2013) 024605

C. Schmitt et al, NPA430 (1984) A. Bail, PRC84 (2011)

Evolution of yields with fissioning system

D. Ramos, USC PhD

Evolution of yields with E*

D. Ramos, USC PhD

Vx_CM (cm/ns)-1 -0.5 0 0.5 1

Vy_C

M (c

m/n

s)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Z=40

Z=58

Reconstruction of the vecolity vector in the reference frame of the fissioning system

250Cf (θ=φ=0) VFS= reaction kinematics 240Pu(θ,φVFS) measured in SPIDER assuming a direct (two-body) reaction.

(cm

/ns)

fiss

V

Neutron number N

45 50 55 60 65 701.4

1.6

Z=35

55 60 65 70 75 801.3

1.4

1.5

Z=42

65 70 75 80 85 901.1

1.2

1.3

Z=49

75 80 85 90 950.9

1

1.1Z=56

45 50 55 60 65 701.4

1.6

Z=3655 60 65 70 75 801.2

1.3

1.4

Z=4365 70 75 80 85 901.1

1.2

1.3

Z=50

75 80 85 90 950.9

1

1.1

Z=57

45 50 55 60 65 701.4

1.6Z=37

55 60 65 70 75 801.2

1.3

1.4

Z=4465 70 75 80 85 901

1.2

Z=5175 80 85 90 950.9

1

1.1

Z=58

45 50 55 60 65 701.4

1.6

Z=38

55 60 65 70 75 801.2

1.3

1.4

Z=45

65 70 75 80 85 901

1.2

Z=5275 80 85 90 950.8

0.9

1

Z=59

45 50 55 60 65 701.3

1.4

1.5

Z=3955 60 65 70 75 801.2

1.3

1.4

Z=46

65 70 75 80 85 901

1.2

Z=53

75 80 85 90 950.8

0.9

1 Z=60

45 50 55 60 65 701.3

1.4

1.5

Z=4055 60 65 70 75 801.1

1.2

1.3

Z=4765 70 75 80 85 901

1.2

Z=54

75 80 85 90 950.8

0.9

1Z=61

45 50 55 60 65 701.3

1.4

1.5Z=41

55 60 65 70 75 801.1

1.2

1.3

Z=4865 70 75 80 85 900.9

1

1.1

Z=5575 80 85 90 950.8

0.9

1

Z=62

Reconstruction of the vecolity vector in the reference frame of the fissioning system

2

slowing-down of the fission fragments into the target hasbeen taken into account, whereas it was considered asnegligible previously. For each isotope, the velocity mea-sured in the laboratory is then corrected for the energy-loss following the prescription of [9], in which the di↵er-ent parameters are adjusted by means of LISE++ sim-ulations [10]. In addition, the velocity distributions ofeach fragment have been corrected for transmission cuts(angle and ionic charge states) that modified slightly themean value of the distribution. The resulting velocityvectors are transformed into the reference frame of thefissioning system. The resolution on the resulting fissionvelocities is depending on the resolution on the velocityand the angle in the laboratory reference frame, and thebeam-energy straggling. Considering a resolution of 0.4%on the velocity measurement and an angular resolutionof 5 mrad [], the resolution on the resulting fission ve-locity was estimated better than 2%. The resulting firstand second momentum of the fission velocity distributionV (A,Z) and �V (A,Z) are displayed for each fragmentisotopicaly identified, in figures 1 and 2, for 240Pu and250Cf fissioning systems , respectively.

The average velocity < V > (Z) for each atomic num-ber Z and its average standard deviation are defined as:

< V > (Z) =

PA

Y (A,Z)V (Z,A)PA

Y (A,Z)

< �V > (Z) =

PA

Y (A,Z)�V (Z,A)PA

Y (A,Z)

(1)

They are displayed in figure 3 and 4 for both sys-tems. The average velocity < V > (Z) is compared tothe liquid-drop model prediction of the fission kinemat-ics [11], with constant deformation and neck parameters.Following this prescription, the total kinetic energy TKEat scission is given by:

TKE = 1.44Z1Z2

D(2)

where D is the distance between the charged centroidsof both fragments A⇤

1 and A⇤2, and may be written as a

function of the fragment deformation parameters �1 and�2 and d the distance between them:

D = r0(A⇤11/3(1 +

2

3�1) +A⇤

21/3(1 +

2

3�2)) + d, (3)

where r0 is the nucleon radius. The velocity of one frag-ment is deduced from the mass and momentum conser-vation. The average mass of both fragments at scission,before neutron evaporation, obtained experimentatly asdescribed in the following section, is considered. For abetter agreement of the Wilkins prescription with thepresent data, the neck parameter d needed to be in-creased from 2 to 2.7 fm for 250Cf fission and 2.5 fmfor 240Pu fission. This increase could be a result of thereaction mechanism used in the present work, inducinghigher angular momentum than in the proton-induced orspontaneous fission considered by Wilkins. With respect

(cm

/ns)

fis

sV

Neutron number N

45 50 55 60 65 701.4

1.6Z=34

55 60 65 70 75 801.3

1.4

1.5

Z=41

65 70 75 80 85 901

1.2

Z=4875 80 85 90 950.9

1

1.1

Z=55

45 50 55 60 65 701.4

1.6

Z=35

55 60 65 70 75 801.2

1.3

1.4

Z=42

65 70 75 80 85 901

1.2Z=49

75 80 85 90 950.8

0.9

1 Z=56

45 50 55 60 65 701.4

1.6

Z=36

55 60 65 70 75 801.2

1.3

1.4

Z=4365 70 75 80 85 901

1.2Z=50

75 80 85 90 950.8

0.9

1

Z=57

45 50 55 60 65 701.3

1.4

1.5

Z=37

55 60 65 70 75 801.2

1.3

1.4

Z=44

65 70 75 80 85 901

1.2

Z=51

75 80 85 90 950.8

0.9

1Z=58

45 50 55 60 65 701.3

1.4

1.5

Z=38

55 60 65 70 75 801.2

1.3

1.4

Z=45

65 70 75 80 85 901

1.2

Z=52

75 80 85 90 950.8

0.9

1

Z=59

45 50 55 60 65 701.3

1.4

1.5

Z=3955 60 65 70 75 801.1

1.2

1.3Z=46

65 70 75 80 85 900.9

1

1.1

Z=5375 80 85 90 950.8

0.9

1

Z=60

45 50 55 60 65 701.3

1.4

1.5

Z=40

55 60 65 70 75 801.1

1.2

1.3Z=47

65 70 75 80 85 900.9

1

1.1Z=54

FIG. 1. Mean values of the fission velocity spectra as a func-tion of the neutron number of the isotopes produced in thefission of 240Pu, for each atomic number. The error bars showthe second momentum of the velocity spectra.

to previous work [5], a better agreement with the theo-retical expectation is reached, as the correction for theenergy loss in the target is now taken into account. Infigure 3, some deviations around Z ⇠ 52 and Z ⇠ 42 withrespect to the liquid-drop model can be observed. Theyare the signature of the presence of shell e↵ects in thedeformation configuration, as will be discussed further.

The observed standard deviation is the quadratic sumof the experimental error, the physical distribution ofscission configurations (resulting from an ensemble of dif-ferent neck or deformations for the same split), and ve-locity spread due to neutron evaporation. In both sys-tems the standard deviation �V of the velocity is decreas-ing with increasing Z, showing that the di↵erent fluctu-ations in the scission configuration and subsequent neu-tron evaporation are less and less influencing the strag-gling on the fragment momentum as the fission massincreasses. However, the standard deviation observed

Average fission velocities

Z35 40 45 50 55 60

<V>

(cm

/ns)

0.8

1

1.2

1.4

35 40 45 50 55 600.99

1

1.01

Z35 40 45 50 55 60

> (c

m/n

s)Vσ<

0.04

0.05

0.06

0.07

Z

Z

Z35 40 45 50 55 60

<V>

(cm

/ns)

0.8

1

1.2

1.4

35 40 45 50 55 60

1

1.05

Z35 40 45 50 55 60

> (c

m/n

s)Vσ<

0.04

0.05

0.06

0.07

0.08

Z

Z

2



< V > (Z) =

PA

Y (A,Z)V (Z,A)PA

Y (A,Z)

< �V > (Z) =

PA

Y (A,Z)�V (Z,A)PA

Y (A,Z)

(1)


TKE = 1.44Z1Z2

D(2)




D = r0(A⇤11/3(1 +

2

3�1) +A⇤

21/3(1 +

2

3�2)) + d, (3)


(cm

/ns)

fiss

V

Neutron number N

45 50 55 60 65 701.4

1.6Z=34

55 60 65 70 75 801.3

1.4

1.5

Z=41

65 70 75 80 85 901

1.2

Z=4875 80 85 90 950.9

1

1.1

Z=55

45 50 55 60 65 701.4

1.6

Z=35

55 60 65 70 75 801.2

1.3

1.4

Z=42

65 70 75 80 85 901

1.2Z=49

75 80 85 90 950.8

0.9

1 Z=56

45 50 55 60 65 701.4

1.6

Z=36

55 60 65 70 75 801.2

1.3

1.4

Z=4365 70 75 80 85 901

1.2Z=50

75 80 85 90 950.8

0.9

1

Z=57

45 50 55 60 65 701.3

1.4

1.5

Z=37

55 60 65 70 75 801.2

1.3

1.4

Z=44

65 70 75 80 85 901

1.2

Z=51

75 80 85 90 950.8

0.9

1Z=58

45 50 55 60 65 701.3

1.4

1.5

Z=38

55 60 65 70 75 801.2

1.3

1.4

Z=45

65 70 75 80 85 901

1.2

Z=52

75 80 85 90 950.8

0.9

1

Z=59

45 50 55 60 65 701.3

1.4

1.5

Z=3955 60 65 70 75 801.1

1.2

1.3Z=46

65 70 75 80 85 900.9

1

1.1

Z=5375 80 85 90 950.8

0.9

1

Z=60

45 50 55 60 65 701.3

1.4

1.5

Z=40

55 60 65 70 75 801.1

1.2

1.3Z=47

65 70 75 80 85 900.9

1

1.1Z=54




2



< V > (Z) =

PA

Y (A,Z)V (Z,A)PA

Y (A,Z)

< �V > (Z) =

PA

Y (A,Z)�V (Z,A)PA

Y (A,Z)

(1)


TKE = 1.44Z1Z2

D(2)




D = r0(A⇤11/3(1 +

2

3�1) +A⇤

21/3(1 +

2

3�2)) + d, (3)


(cm

/ns)

fiss

V

Neutron number N

45 50 55 60 65 701.4

1.6Z=34

55 60 65 70 75 801.3

1.4

1.5

Z=41

65 70 75 80 85 901

1.2

Z=4875 80 85 90 950.9

1

1.1

Z=55

45 50 55 60 65 701.4

1.6

Z=35

55 60 65 70 75 801.2

1.3

1.4

Z=42

65 70 75 80 85 901

1.2Z=49

75 80 85 90 950.8

0.9

1 Z=56

45 50 55 60 65 701.4

1.6

Z=36

55 60 65 70 75 801.2

1.3

1.4

Z=4365 70 75 80 85 901

1.2Z=50

75 80 85 90 950.8

0.9

1

Z=57

45 50 55 60 65 701.3

1.4

1.5

Z=37

55 60 65 70 75 801.2

1.3

1.4

Z=44

65 70 75 80 85 901

1.2

Z=51

75 80 85 90 950.8

0.9

1Z=58

45 50 55 60 65 701.3

1.4

1.5

Z=38

55 60 65 70 75 801.2

1.3

1.4

Z=45

65 70 75 80 85 901

1.2

Z=52

75 80 85 90 950.8

0.9

1

Z=59

45 50 55 60 65 701.3

1.4

1.5

Z=3955 60 65 70 75 801.1

1.2

1.3Z=46

65 70 75 80 85 900.9

1

1.1

Z=5375 80 85 90 950.8

0.9

1

Z=60

45 50 55 60 65 701.3

1.4

1.5

Z=40

55 60 65 70 75 801.1

1.2

1.3Z=47

65 70 75 80 85 900.9

1

1.1Z=54




d=2.5 fm for 240Pu (2p transfer, E*=9 MeV) d=2.7 fm for 250Cf (fusion, E*=45 MeV) Β1=β2=0.6

A very complete and constraining set of data

Z35 40 45 50 55 60

TKE

(MeV

)170

172

174

176

178

180

182

Z35 40 45 50 55 60

TKE

(MeV

)

160

165

170

175

180

Deformation at scission !!

240Pu E*~9MeV 250Cf E*~45MeV

M. Caamaño, F. Farget et al.,

Scission fragment characterization !!

3

(cm

/ns)

fiss

V

Neutron number N

45 50 55 60 65 701.4

1.6

Z=35

55 60 65 70 75 801.3

1.4

1.5

Z=42

65 70 75 80 85 901.1

1.2

1.3

Z=49

75 80 85 90 950.9

1

1.1Z=56

45 50 55 60 65 701.4

1.6

Z=3655 60 65 70 75 801.2

1.3

1.4

Z=4365 70 75 80 85 901.1

1.2

1.3

Z=50

75 80 85 90 950.9

1

1.1

Z=57

45 50 55 60 65 701.4

1.6

Z=37

55 60 65 70 75 801.2

1.3

1.4

Z=4465 70 75 80 85 901

1.2

Z=5175 80 85 90 950.9

1

1.1

Z=58

45 50 55 60 65 701.4

1.6

Z=38

55 60 65 70 75 801.2

1.3

1.4

Z=45

65 70 75 80 85 901

1.2

Z=5275 80 85 90 950.8

0.9

1

Z=59

45 50 55 60 65 701.3

1.4

1.5

Z=3955 60 65 70 75 801.2

1.3

1.4

Z=46

65 70 75 80 85 901

1.2

Z=53

75 80 85 90 950.8

0.9

1 Z=60

45 50 55 60 65 701.3

1.4

1.5

Z=4055 60 65 70 75 801.1

1.2

1.3

Z=4765 70 75 80 85 901

1.2

Z=54

75 80 85 90 950.8

0.9

1Z=61

45 50 55 60 65 701.3

1.4

1.5

Z=41

55 60 65 70 75 801.1

1.2

1.3

Z=4865 70 75 80 85 900.9

1

1.1

Z=5575 80 85 90 950.8

0.9

1

Z=62

FIG. 2. Mean values of the fission velocity spectra as a func-tion of the neutron number of the di↵erent isotopes producedin the fission of 250Cf, for each atomic number. The error barsshow the second momentum of the velocity spectra.

in the fission of Cf is larger, reflecting a higher excita-tion energy of the compound nucleus, and therefore alarger number of evaporated neutrons as a larger ensem-ble of scission configurations. The standard deviationsobserved in the fission of Pu show fluctuations, revealingthe low statistics observed mainly in the symmetric split.

III. RECONSTRUCTION OF THE SCISSIONFRAGMENT MASS

The fission velocity is reflecting important propertiesof the scission configuration, such as deformation andmasses of the nascent fragments. Due to the momentumconservation, the ratio of the two fragment velocities isequal to the reverse ratio of the initial masses:

V1

V2=

A⇤2

A⇤1

(4)

Z35 40 45 50 55 60

<V>

(cm

/ns)

0.8

1

1.2

1.4

35 40 45 50 55 60

1

1.05

Z35 40 45 50 55 60

> (c

m/n

s)Vσ<

0.04

0.05

0.06

0.07

0.08

FIG. 3. (Color online) Upper panel: average fission velocities< V > (Z) as a function of the fragment atomic numberZ produced in the fission of 240Pu. It is compared to theWilkins prescription of the fission kinematics (red line). Theneck parameter d has been increased from 2 to 2.5 fm. Bottompanel: the average standard deviation of the fission-velocityspectra. See equation 1 and text for details.

Z35 40 45 50 55 60

<V>

(cm

/ns)

0.8

1

1.2

1.4

35 40 45 50 55 600.99

1

1.01

Z35 40 45 50 55 60

> (c

m/n

s)Vσ<

0.04

0.05

0.06

0.07

FIG. 4. (Color online) Upper panel: average fission velocities< V > (Z) as a function of the fragment atomic number Zproduced in the fission of 250Cf. It is compared to the Wilkinsprescription of the fission kinematics (red line). Bottom panel:average standard deviation of the fission-velocity spectra. Seeequation 1 and text for details.

4

In the present experiment, the velocity of only one frag-ment is measured, however over the complete fragmentproduction, as demonstrated in the preceeding section.In both investigated systems the excitation energy is notenough to allow for proton evaporation, therefore for onefissioning system of definite atomic number ZFS one frag-ment atomic number Z1 is associated to the complemen-tary atomic number Z2 = ZFS � Z1. In the case of240Pu fissioning system, with an average excitation en-ergy of 9 MeV, no pre-scission neutron evaporation isconsidered. In the case of 250Cf, the excitation energy issu�cient to allow for neutron evaporation before fission-ing. The neutron-evaporation probability depends on theangular momentum induced in the reaction. In similarreaction with 16O beam on 238U, angular anisotropy hasbeen used to determine the root-mean-squared angularmomentum induced in the reaction at an energy close tothe Coulomb barrier [12]. A value of 24h̄ was determinedfor this reaction. Calculations based on the Bass modeli-sation [13] of which the parameter are slightly adjustedto reproduce the experimental data of [12], an estimationof 20h̄ is done in the case of the present experiment. Thisangular momentum and excitation energy lead to an av-erage fissioning system of mass AFS = 249.5, consideringthe di↵erent fission probabilities obtained from GEF [14]predictions. Indeed, from this simulation code, the firstchance fission is supposed to happen in more than 64% ofthe cases. The error estimated on the fissioning nucleusaverage mass is restricted to 0.3 mass units, consideringthe small possible variations on angular momentum andthe limited excitation energy. It is then possible to asso-ciate to both fission-fragment atomic number the averagefission velocities< V1 > and< V2 > from equation 1, anddeduce the average initial masses < A⇤

1 > and < A⇤2 >

using the momentum and mass conservation:

< A⇤1 >= AFS

<V2><V1>

< A⇤2 >= AFS� < A⇤

1 >(5)

The resulting average neutron excess of the scissionfragments is displayed in blue open circles in figures 5and 6 for both fissioning systems. The neutron excessdefined as <A>(Z)�Z

Z =< N > /Z, or charge polarisationof the fission fragments, is chosen, as it shows a moreemphasized structure compared to the simple < A >(Z) mass information, which is increasing steadily withZ. The error bars displayed comprise the uncertainty onthe fission velocity measurement and on the fissioning-nucleus mass, as the statistical source of uncertainty. Itis compared to the post-evaporation neutron-excess ofthe fragments, displayed in black full circles, deducedfrom the average post-evaporation mass measured in theexperiment:

< A > (Z) =

PA AY (A,Z)PA Y (A,Z)

(6)

The neutron excess of the scission fragments of 240Pu

(open circles) show a step behaviour, with a sudden in-crease around Z = 50. The step is maintened up toZ = 54 and above. The subsequent neutron evapora-tion does not attenuate the step structure, which is alsoobserved for the post neutron-evaporation fragments.In contrary, the neutron excess of the scission frag-

ments of 250Cf shows a smooth behaviour and a steady in-crease with increasing Z, while neutron evaporation mod-ifies significantly the trend. The fragments observed af-ter neutron-evaporation show a constant neutron-excess,with no charge polarisation.The larger error bars observed in the fragments of

240Pu reflect the lower statistics in the population of thisfissioning system, due to the lower transfer cross sectioncompared to the fusion cross section, by about a factor50 [8, 15].

Z35 40 45 50 55 60

<N>/Z

1.45

1.5

1.55

1.6

FIG. 5. (Color online) Average neutron excess < N > /Z ofthe fragments produced in the fission of 240Pu as a function oftheir atomic number Z, pre and post-neutron evaporation, inblue open and black full circles, respectively. It is comparedto the estimation of GEF code with the same color code,open and full triangles for pre and post-neutron evaporationfragments. The red dashed-line corresponds to the predictionsof the liquid-drop scission-point model.

These experimental results are compared to GEF cal-culations [14], displayed in both figures as open and fulltriangles, for the neutron excess of scission and final frag-ments, respectively. The structures appearing in the neu-tron excess of the 240Pu scission and final fragments arewell reproduced by the GEF code. In contrast, more de-viations with respect to the experimental data appearfor the calculation of the fragments of 250Cf. This com-pound nucleus is populated at an excitation energy of45 MeV, and therefore it is expected that the contribu-tions of the single-particle structure of the nucleus havedisapeared [16]. Indeed, GEF calculations show a simi-lar steady increase of the neutron excess with the atomicnumber of the scission fragments, with a slight di↵erentslope that will be discussed in section VI. However, thesecalculations cannot reproduce the experimental constantvalue of the neutron excess of the final fragments as afunction of their atomic number. Indeed, the light fis-

Z35 40 45 50 55 60

<N>/Z

1.45

1.5

1.55

1.6

Z35 40 45 50 55 60

<N>/Z

1.4

1.45

1.5

1.55

1.6

scission

fission

M. Caamaño, F. Farget et al.,


Momentum conservation

2

44 46 48 50 52 54 56 581.41.451.51.551.61.65

Z_3546 48 50 52 54 56 58 60

1.41.451.51.551.61.65

Z_3646 48 50 52 54 56 58 60 621.35

1.41.451.51.551.6

Z_37

48 50 52 54 56 58 60 62 641.351.41.451.51.551.6

Z_3848 50 52 54 56 58 60 62 64 661.3

1.351.41.451.51.55

Z_3950 52 54 56 58 60 62 64 66 68

1.31.351.41.451.51.55

Z_40

52 54 56 58 60 62 64 66 68 701.251.31.351.41.451.5

Z_4152 54 56 58 60 62 64 66 68 70 72

1.251.31.351.41.451.5

Z_4254 56 58 60 62 64 66 68 70 721.2

1.251.31.351.41.45

Z_43

55 60 65 70 751.21.251.31.351.41.45

Z_4456 58 60 62 64 66 68 70 72 74 76

1.151.21.251.31.351.4

Z_4558 60 62 64 66 68 70 72 74 76

1.151.21.251.31.351.4

Z_46

60 65 70 75 801.1

1.151.21.251.31.35

Z_4760 65 70 75 80

1.11.151.21.251.31.35

Z_4862 64 66 68 70 72 74 76 78 80 82

1.051.1

1.151.21.251.3

Z_49

64 66 68 70 72 74 76 78 80 82 84

1.051.1

1.151.21.251.3

Z_5065 70 75 80 85

11.051.1

1.151.21.25

Z_5165 70 75 80 85

11.051.1

1.151.21.25

Z_52

68 70 72 74 76 78 80 82 84 86 880.95

11.051.1

1.151.2

Z_5370 75 80 85 90

0.951

1.051.1

1.151.2

Z_5470 75 80 85 90

0.90.95

11.051.1

1.15

Z_55

74 76 78 80 82 84 86 88 90 92

0.90.95

11.051.1

1.15

Z_5674 76 78 80 82 84 86 88 90 92 94

0.850.90.95

11.051.1

Z_5775 80 85 90 95

0.850.90.95

11.051.1

Z_58

78 80 82 84 86 88 90 92 940.80.850.90.95

11.05

Z_5978 80 82 84 86 88 90 92 94 96

0.80.850.90.95

11.05

Z_6080 82 84 86 88 90 92 94 96 980.75

0.80.850.90.95

1

Z_61

82 84 86 88 90 92 94 96 980.750.80.850.90.95

1

Z_6286 88 90 92 94 96 98

0.750.80.850.90.95

1

Z_63

FIG. 2. Average fission velocities as a function of the massesof the di↵erent isotopes produced in the fission of 250Cf, foreach atomic number.

before neutron evaporation, obtained experimentatly asdescribed below, is considered in the estimation of the fis-sion velocities from the liquid-drop model. For a betteragreement of the prescription of Wilkins with the presentdata, a neck parameter of 2.5 fm needs to be considered.With respect to previous work [1], a better agreementwith the theoretical estimation is reached, as the correc-tion for the energy loss in the target has been taken intoaccount.

The increase of the neck parameter with respect toWilkins prescription could be a result of the reactionmechanism, inducing higher angular momentum than inthe proton-induced or spontaneous fission considered byWilkins. In figure 3, some deviations with respect to theliquid-drop model can be observed. They are the signa-ture of the presence of shell e↵ects as will be discussedfurther.

In both systems the standard deviation is decreasing

Z35 40 45 50 55 60

<V>

(cm

/ns)

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Z35 40 45 50 55 60

RM

S (c

m/n

s)

0.02

0.04

0.06

0.08

0.1

FIG. 3. Average fission velocities < V > (Z) as a functionof the fragment atomic number Z produced in the fission of240Pu. It is compared to the Wilkins prescription of the fis-sion kinematics (red line). The neck parameter d has beenincreased from 2 to 2.5 fm.The average standard deviation isdisplayed in the bottom panel. See equation 1 and text fordetails.

with increasing Z, showing that the di↵erent fluctua-tions in the scission configuration and subsequent neu-tron evaporation are less and less influencing stragglingon the fragment momentum as the fission mass incre-asses. The standard deviation for the Cf fissioning sys-tem is larger, reflecting a higher excitation energy. some

arguments?

RECONSTRUCTION OF THE SCISSIONFRAGMENT MASS

The velocity is reflecting important properties of thescission configuration, such as deformation and massesof the nascent fragments. Due to the momentum conser-vation, the ratio of the velocities is equal to the reverseratio of the initial masses:

V1

V2=

A⇤2

A⇤1

(2)

In the present experiment, only one velocity is measured,however the complete fragment production is covered, asdemonstrated in the preceeding section. In both inves-tigated systems the excitation energy is not enough toallow for proton evaporation, therefore for one fission-ing system of definite atomic number ZFS one fragmentatomic number Z1 is associated to the complementaryatomic number Z2 = ZFS � Z1. In the case of 240Pu

Charge and mass conservation

Scission configuration investigation: 240Pu E*=9MeV

Scission point model: minimisation of the total energy

Liquid-drop energy Myers & Swiatecki, Lysekil, Ark. Phys. 36 (1967) 343

Scission configuration investigation: 250Cf E*=45MeV Scission point model: minimisation of the total energy


*0.4

*0.95

Persistence of SE at E*=45 MeV !!

Persistence of Shell Effects at E* = 45 MeV !!

Shell effects are expected to vanish exponentialy with E*

Scission configuration investigation: 250Cf E*=45MeV Scission point model: minimisation of the total energy


*2./3

Reduction of the Symmetry energy term to reproduce the slope

Diminution of symmetry energy with deformation ?

GAIDAROV, ANTONOV, SARRIGUREN, AND DE GUERRA PHYSICAL REVIEW C 85, 064319 (2012)

These increases are only slightly dependent on whether anoblate or a prolate shape is considered. A similar behavior hasbeen obtained from Gogny-D1S-HFB calculations performedin Ref. [50]. A satisfactory agreement with the experimentalisotope shifts is observed in Fig. 1(c) that provides a goodstarting point to study further quantities such as the symmetryenergy and related characteristics of deformed nuclei withinour theoretical method.

Next, an illustration of a possible correlation of the neutronskin thickness !R with the s and p0 parameters, extracted fromthe density dependence of the symmetry energy around thesaturation density for the Kr isotopic chain, is given in Fig. 2.The symmetry energy and the pressure are calculated withinthe CDFM according to Eqs. (10) and (11) by using the weightfunctions (9) calculated from the self-consistent densities inEq. (21). The differences between the neutron and proton rmsradii of these isotopes [Eq. (29)] are obtained from HF + BCScalculations using four different Skyrme forces, SLy4, SGII,Sk3, and LNS. It can be seen from Fig. 2 that there exists anapproximate linear correlation between !R and s for the even-even Kr isotopes with A = 82–96. Similarly to the behavior of!R vs s dependence for the cases of Ni and Sn isotopes [24],we observe a smooth growth of the symmetry energy up to thesemimagic nucleus 86Kr (N = 50) and then a linear decreaseof s while the neutron skin thickness of the isotopes increases.This linear tendency expressed for Kr isotopes with A > 86is similar for the cases of both oblate and prolate deformedshapes. We note that all Skyrme parametrizations used in thecalculations reveal similar behavior; in particular, the averageslope of !R for various forces is almost the same.

In addition, one can see from Fig. 2 a stronger deviationbetween the results for oblate and prolate shape of Kr isotopesin the case of SGII parametrization when displaying thecorrelation between !R and s. This is valid also for thecorrelation between !R and p0, where more distinguishableresults for both types of deformation are present. The neutronskin thickness !R for Kr isotopes correlates with p0 almostlinearly, as in the symmetry-energy case, with an inflection-point transition at the semimagic 86Kr nucleus. In addition,one can see also from Fig. 2 that the calculated values forp0 are smaller in the case of LNS and SLy4 forces than forthe other two Skyrme parameter sets. In general, we wouldlike to note that the behavior of deformed Kr isotopes shownin Fig. 2 is comparable with the one found for the sphericalNi and Sn isotopes having a magic proton number that wediscussed in Ref. [24]. The small differences just indicate thatstability patterns are less regular within isotopic chains with anonmagic proton number.

For more complete study, we also consider in our work theextremely neutron-rich Kr isotopes (A = 96–120). The resultsfor the symmetry energy s as a function of the mass number Afor the whole Kr isotopic chain (A = 82–120) are presentedin Fig. 3. We observe peaks of the symmetry energy at specificKr isotopes, namely at semimagic 86Kr (N = 50) and 118Kr(N = 82) nuclei. In addition, a flat area is found surrounded bytransitional regions A = 88–96 and A = 110–116. Also, theSGII and Sk3 forces yield values of s comparable with eachother that lie between the corresponding symmetry energyvalues when using SLy4 and LNS sets. The specific nature

FIG. 3. (Color online) The symmetry energies s for Kr isotopes(A = 82–120) calculated with SLy4, SGII, Sk3, and LNS forces.

of the LNS force [48] (not being fitted to finite nuclei) leadsto larger values of s (and to a larger size of the neutron skinthickness, as is seen from Fig. 2) with respect to the results withother three forces. Although the values of s slightly vary withinthe Kr isotopic chain when using different Skyrme forces, thecurves presented in Fig. 3 exhibit the same trend.

The results shown in Fig. 3 are closely related to the evolu-tion of the quadrupole parameter " =

!#/5 Q/(A!r2"1/2) (Q

being the mass quadrupole moment and !r2"1/2 the nucleusrms radius) as a function of the mass number A that ispresented in Fig. 4. First, one can see from Fig. 4 that thesemimagic A = 86 and A = 118 Kr isotopes are spherical,while the open-shell Kr isotopes within this chain possesstwo equilibrium shapes, oblate and prolate. In the case ofopen-shell isotopes, the oblate and prolate minima are veryclose in energy and the energy difference is always less than

84 88 92 96 100 104 108 112 116 120A

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

!

Kr SLy4

FIG. 4. (Color online) The quadrupole parameter " as a functionof the mass number A for the even-even Kr isotopes (A = 82–120)in the case of the SLy4 force.

064319-6

GAIDAROV, ANTONOV, SARRIGUREN, AND DE GUERRA PHYSICAL REVIEW C 85, 064319 (2012)

These increases are only slightly dependent on whether anoblate or a prolate shape is considered. A similar behavior hasbeen obtained from Gogny-D1S-HFB calculations performedin Ref. [50]. A satisfactory agreement with the experimentalisotope shifts is observed in Fig. 1(c) that provides a goodstarting point to study further quantities such as the symmetryenergy and related characteristics of deformed nuclei withinour theoretical method.

Next, an illustration of a possible correlation of the neutronskin thickness !R with the s and p0 parameters, extracted fromthe density dependence of the symmetry energy around thesaturation density for the Kr isotopic chain, is given in Fig. 2.The symmetry energy and the pressure are calculated withinthe CDFM according to Eqs. (10) and (11) by using the weightfunctions (9) calculated from the self-consistent densities inEq. (21). The differences between the neutron and proton rmsradii of these isotopes [Eq. (29)] are obtained from HF + BCScalculations using four different Skyrme forces, SLy4, SGII,Sk3, and LNS. It can be seen from Fig. 2 that there exists anapproximate linear correlation between !R and s for the even-even Kr isotopes with A = 82–96. Similarly to the behavior of!R vs s dependence for the cases of Ni and Sn isotopes [24],we observe a smooth growth of the symmetry energy up to thesemimagic nucleus 86Kr (N = 50) and then a linear decreaseof s while the neutron skin thickness of the isotopes increases.This linear tendency expressed for Kr isotopes with A > 86is similar for the cases of both oblate and prolate deformedshapes. We note that all Skyrme parametrizations used in thecalculations reveal similar behavior; in particular, the averageslope of !R for various forces is almost the same.

In addition, one can see from Fig. 2 a stronger deviationbetween the results for oblate and prolate shape of Kr isotopesin the case of SGII parametrization when displaying thecorrelation between !R and s. This is valid also for thecorrelation between !R and p0, where more distinguishableresults for both types of deformation are present. The neutronskin thickness !R for Kr isotopes correlates with p0 almostlinearly, as in the symmetry-energy case, with an inflection-point transition at the semimagic 86Kr nucleus. In addition,one can see also from Fig. 2 that the calculated values forp0 are smaller in the case of LNS and SLy4 forces than forthe other two Skyrme parameter sets. In general, we wouldlike to note that the behavior of deformed Kr isotopes shownin Fig. 2 is comparable with the one found for the sphericalNi and Sn isotopes having a magic proton number that wediscussed in Ref. [24]. The small differences just indicate thatstability patterns are less regular within isotopic chains with anonmagic proton number.

For more complete study, we also consider in our work theextremely neutron-rich Kr isotopes (A = 96–120). The resultsfor the symmetry energy s as a function of the mass number Afor the whole Kr isotopic chain (A = 82–120) are presentedin Fig. 3. We observe peaks of the symmetry energy at specificKr isotopes, namely at semimagic 86Kr (N = 50) and 118Kr(N = 82) nuclei. In addition, a flat area is found surrounded bytransitional regions A = 88–96 and A = 110–116. Also, theSGII and Sk3 forces yield values of s comparable with eachother that lie between the corresponding symmetry energyvalues when using SLy4 and LNS sets. The specific nature

FIG. 3. (Color online) The symmetry energies s for Kr isotopes(A = 82–120) calculated with SLy4, SGII, Sk3, and LNS forces.

of the LNS force [48] (not being fitted to finite nuclei) leadsto larger values of s (and to a larger size of the neutron skinthickness, as is seen from Fig. 2) with respect to the results withother three forces. Although the values of s slightly vary withinthe Kr isotopic chain when using different Skyrme forces, thecurves presented in Fig. 3 exhibit the same trend.

The results shown in Fig. 3 are closely related to the evolu-tion of the quadrupole parameter " =

!#/5 Q/(A!r2"1/2) (Q

being the mass quadrupole moment and !r2"1/2 the nucleusrms radius) as a function of the mass number A that ispresented in Fig. 4. First, one can see from Fig. 4 that thesemimagic A = 86 and A = 118 Kr isotopes are spherical,while the open-shell Kr isotopes within this chain possesstwo equilibrium shapes, oblate and prolate. In the case ofopen-shell isotopes, the oblate and prolate minima are veryclose in energy and the energy difference is always less than

84 88 92 96 100 104 108 112 116 120A

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

!

Kr SLy4

FIG. 4. (Color online) The quadrupole parameter " as a functionof the mass number A for the even-even Kr isotopes (A = 82–120)in the case of the SLy4 force.

064319-6

Gaidarov et al., PRC 85 (2012) 064319

Diminution of symmetry energy with isospin?

Ning Wang PRC 81 (2010)067302

Post-scission neutron evaporation

Z35 40 45 50 55 60

>(Z)

ν<

012345678

Z35 40 45 50 55 60

>(Z)

ν<

1

2

3

4

5

6

7

8

D. H

inde

et a

l, P

RC

45(1

992)


Summary and outlook

•  Transfer-induced fission in inverse kinematics coupled to the spectrometer VAMOS allows to –  Investigate A ten of fissioning actinides, heavier than 238U –  With E* ranging from few MeV above the fission barrier to 45 MeV –  For each of the fissioning systems the isotopic fission-fragment

distributions are avaible –  With kinematics properties of the fission fragments it is possible to

reconstruct the properties of the fragments at scission •  Their TKE •  Their average neutron excess <N>/Z

–  The present results show the importance to consider polarisation in the emergence of the fragments

–  Further developments in the description of the scission fragments (evolution of binding energy with E*, deformation,…) are needed !!

–  Sharing of E* seems to behave differently than described by a thermal equilibrium of a Fermi gas.

transfer-induced fission in inverse kinematics: from...

Documents