Light fission-fragment mass distribution from the reaction 251Cf(nth, f )

transcript

Nuclear Physics A 791 (2007) 1–23

Light fission-fragment mass distributionfrom the reaction 251Cf(nth, f )

E. Birgersson a,b, S. Oberstedt a,∗, A. Oberstedt b, F.-J. Hambsch a,D. Rochman c, I. Tsekhanovich d,1, S. Raman e,�

a EC-JRC Institute for Reference Materials and Measurements (IRMM), B-2440 Geel, Belgiumb Institutionen för Naturvetenskap, Örebro Universitet, S-70182 Örebro, Sweden

c Brookhaven National Laboratory, National Nuclear Data Center, Upton, NY 11973-5000, USAd Institut Laue-Langevin, 6 rue Jules Horowitz, F-38042 Grenoble, France

e Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

Received 28 September 2006; received in revised form 9 March 2007; accepted 3 April 2007

Available online 29 April 2007

Abstract

For mass numbers A = 80 to 124 the recoil mass spectrometer LOHENGRIN of the Institute Laue-Langevin in Grenoble was used to measure with high resolution the light fission-fragment mass yields andkinetic energy distributions from thermal-neutron induced fission of 252Cf* for the first time, using 251Cfas target material. The obtained mean light fragment mass 〈AL〉 = (107 ± 2) and the corresponding meankinetic energy 〈Ek,L〉 = (103 ± 2) MeV are within the expected trend. Emission yields around A = 115 areenhanced and the corresponding mean kinetic energy is higher compared to spontaneous fission of 252Cf.This could be explained by the existence of an additional super-deformed fission mode.© 2007 Elsevier B.V. All rights reserved.

PACS: 25.85.Ec; 28.20.-v; 29.30.Ep; 24.75.+i

Keywords: NUCLEAR REACTIONS, 251Cf(nth, f ); Californium-251; Neutron-induced fission; Fission modes; Minoractinides; Transmutation; Fission-fragment spectroscopy

* Corresponding author.E-mail address: stephan.oberstedt@ec.europa.eu (S. Oberstedt).

1 Present address: Schuster Laboratory, The University of Manchester, Manchester M13 9PL, UK.� Deceased.

2 E. Birgersson et al. / Nuclear Physics A 791 (2007) 1–23

1. Introduction

The fission-fragment mass and kinetic energy distributions are key observables when study-ing the fission process. The investigation of their dependence on excitation energy may revealvaluable information about the shape of the nuclear potential energy landscape around the saddlepoint. Deduced fission barrier penetrabilities are important input parameters for reaction cross-section models and evaluation exercises.

In order to transmute the minor actinide waste from the nuclear reactors of today, the cur-rent plan is to make the minor actinides a part of the reactor fuel in the so-called Generation IVnuclear reactors [1] as well as accelerator driven systems (ADS) [2], which are presently underdevelopment. Such facilities operate with a different neutron spectrum compared to traditionalreactors in order to avoid creation of even more minor actinides. Since the number of delayedneutrons, which is an important safety parameter in a nuclear reactor, is determined by the par-ticular mass (and nuclear charge) distribution of a fissioning actinide, good knowledge of thefission fragment characteristics is needed even as a function of excitation energy. Cf-isotopes arethe heaviest minor actinides with the lowest abundance in a nuclear reactor. However, their fis-sion characteristics become important, when modelling fission fragment yields for Cm and Amisotopes.

Moreover, spontaneous fission of 252Cf is considered as a standard reaction in nuclear physicsconcerning prompt neutron multiplicity and neutron evaporation spectrum studies [3] and veryoften used for the characterisation and calibration of neutron and fission-fragment detectors.

The interpretation of fission-fragment properties in terms of so-called fission modes [4] hasbeen successfully applied in the actinide region to describe mass yield and total kinetic energydistributions as a function of incident neutron energy [5–8]. From investigating fission-fragmentcharacteristics as a function of excitation energy the mass-asymmetric standard I (S1) and stan-dard II (S2) modes as well as the mass-symmetric super-long (SL) mode have been used toconsistently describe the mass- and total kinetic energy (TKE) distributions together with the cor-responding distribution widths. In case of spontaneous fission of 252Cf the traditional number offission modes is not sufficient to properly describe the experimentally obtained fission-fragmentdistributions. Additional theoretically obtained fission modes have to be added to the model toachieve a reasonable description of the fission-fragment data [9].

In case of 252Cf the available 251Cf target material does not allow the measurement ofpre-neutron mass and TKE distributions by means of a 2E experiment. Therefore, post-neutron fission-fragment mass yields and mean kinetic energies were measured for the reaction251Cf(nth, f) and compared to data from spontaneous fission of 252Cf, to obtain an indication forfurther fission modes present in 252Cf.

2. Experiment

The experiment has been performed at the recoil mass-separator LOHENGRIN [10] of theInstitute Laue-Langevin in Grenoble (ILL). In LOHENGRIN the separation of fission-fragmentsis done according to their A/q and EL/q ratios, where A, EL and q are the mass number, kineticenergy and ionic charge state of the fission fragments, respectively. The fission fragments are de-tected after separation with an ionisation chamber, which has a segmented anode to serve as a�E–E telescope. In contrast to traditional 2E experiments performed for the reaction 252Cf(SF)with twin ionisation chambers only one fragment per fission may be detected, however, with su-perior mass resolution. Two examples are shown in Fig. 1, where the same mass (here: A = 100)

E. Birgersson et al. / Nuclear Physics A 791 (2007) 1–23 3

Fig. 1. Left: �E–EL-plot of A/q = 100/20 and EL/q = 97/20; right: �E–EL-plot of A/q = 100/23 andEL/q = 97/23.

is detected at the same energy but at different ionic charge states. Its position in the �E–EL-plotdoes not change as it does for the parasitically detected masses. This provides mass identificationand energy calibration of the ionisation chamber.

To determine the yield and average kinetic energy for an individual fission-fragment withmass A, 6–8 different ionic charge states were measured close to the mean kinetic energy and6–8 different energies were measured at approximately the mean ionic charge state. The energyand ionic charge state where the two distributions were measured for each mass were determinedby looking at mean kinetic energy and mean ionic charge state of the neighbouring masses. Thedistributions were then found by fitting Gaussians to these values. Because of the high neutron

Fig. 2. The relative fission activity as a function of irradiation time. The cross-sections are taken from Refs. [11,12] anda neutron flux of 5.4 × 1014 n/cm2/s was used.

flux the monitoring of the change of the target properties was performed by measuring mass 100at ionic charge state 22 at 6–8 energies 3–5 times a day during the whole experiment.

The energy dispersion of the spectrometer is proportional to the fragment energy, and eachmeasurement can be normalised by dividing the obtained number of events by the LOHENGRINenergy, EL, tuned according to the chosen electric and magnetic fields.

To avoid sparks and to make the electric field homogeneous, a formation is performed on theelectric condenser plates once a day as described in Section 3.

2.1. Target properties

The target material was provided by the Oak Ridge National Laboratory (ORNL) and wasprepared at the Institute of Nuclear Chemistry, University of Mainz. The Cf2O3 was electro-deposited on a titanium backing. The active spot had a diameter of 4 mm and a thickness of87.7 µg/cm2. At ILL the target material was covered with a nickel foil with a thickness of0.25 µm or 222.5 µg/cm2. This was done to minimise loss of target material due to sputter-ing when the target is heated up in the reactor. The nickel foil itself is supported by an acryliclayer, which is supposed to evaporate before the measurements start.

The isotopic composition of the target in the beginning of the experiment was 249Cf (18%),250Cf (35%), 251Cf (46%) and 252Cf (1%). The relative number of fission events from each Cf-isotope has been calculated using the thermal neutron flux at the target of 5.4 × 1014 n/cm2/sand fission cross-section data from Refs. [11,12] and is shown in Fig. 2 as a function of time.

3. Data treatment and analysis

The raw data were determined by selecting a “region of interest” (RoI) in the �E–EL-plot,similar to that shown in Fig. 1, using the computer program MPAWIN [13]. The number of eventsin this RoI, N , has to be corrected in several ways. The high neutron flux causes deterioration

of the target material as a function of time. The limited beam time at the LOHENGRIN massspectrometer and the fact that the target material is consumed very rapidly with time is the reasonfor not measuring all possible settings for the yield, Y(A,q,E). This leads to some corrections,because the yield had to be determined using semi-empirical functions. An additional correctionhad to be applied because of observed drifts of the electric field of the mass spectrometer.

3.1. Corrections due to changing target properties

Due to the high neutron flux the target material properties change as a function of irradiationtime. In order to compare measurements performed at different times, the so-called burn-up has tobe taken into account and monitored. This was done by measuring mass A = 100 at ionic chargestate q = 22+ and kinetic energies EL = 80 to 115 MeV in steps of 5 MeV three to five timesper day. A Gaussian fit to one burn-up measurement provides the mean kinetic energy, width andintensity. An example of a burn-up measurement is shown in the upper left part of Fig. 3. Theintensity, mean kinetic energy and width, respectively for the different burn-up measurementsas a function of time are also shown in Fig. 3. The burn-up is described by the sum of twoexponential functions,

Cb100(t) = C1 · exp(− ln 2t1

t) + C2 · exp(− ln 2t2

C1 + C2, (1)

where the fast component, t1, accounts for material losses during initial heating of the target.The slow component, t2, corresponds to regular burn-up of the target material. In Eq. (1) ob-viously the isotopic composition is neglected, but will be taken carefully into consideration inSection 3.6. The measured data are corrected for burn-up by dividing with Cb100(t), the valueof the sum of the two exponential functions, normalised at t = 0. As it may be seen in Fig. 3,the burn-up data show quite some structure, which cannot be explained as being of statisticalnature. Both the mean kinetic energy and the width show the same correlation with the mea-surement time. The observed deviations appear to be strongly linked to the time passed since aformation. A formation is a slow increase of the high voltage on the electric condenser platesand is performed once a day. This is done to make the electric field homogeneous again and toavoid sparks. As a function of time between two formations, the intensity for the lower energiesdecreases more than the higher energies, which causes the mean kinetic energy to increase andthe width to decrease. Therefore, the observed structure must be attributed to drifts of the electricfield in the condenser of LOHENGRIN. This instability has not been reported previously andwas observed because of the frequent burn-up measurements.

Based on the first five burn-up measurements and those performed directly after every day’sformation, functions describing the burn-up and the increase in mean kinetic energy were found.The width of the distribution is assumed to be constant and an average value was calculated fromthe selected burn-up measurements. The half-lives describing the target burn-up, t1 and t2, weredetermined to (0.38 ± 0.01)d and (4.1 ± 0.1)d , respectively. A calculation of the long half-lifeusing fission cross-section data from Refs. [11,12] taking into account all different Cf-isotopesgives a value of 4.7 days.

The energy loss of the fission-fragments in both the target material and the coveringnickel foil is about 6–8 MeV according to calculations performed with the computer programSRIM2003 [14]. In Fig. 3 the measured increase is about 6 MeV. The energy loss occurs mainlyin the nickel foil (approximately 90%) and does not change with time. This leads to the conclu-sion that there must have been more material on the target causing an additional energy loss, but

Fig. 3. Upper left part: Example of a burn-up measurement, A/q = 100/22, described by a Gaussian. Upper right part:The decrease of the fissile material as a function of time is shown. A two-exponential fit is based on the full squares only,which indicate all burn-up measurements performed directly after a formation as well as the first 5 burn-up measurements.The results of the calculated burn-up (see text) is shown as dashed line starting from the intensity at day 4. Lower left:The increase in mean kinetic energy is shown together with a two-exponential fit, again based on the squares only (dashedline). An exponential function with time dependence according to Ref. [15] is also shown (full line). Lower right: Thewidths of the energy distributions. The mean value is calculated using only the full squares. All measurements (opencircles indicates measurements between two successive formations) were used to monitor the electronic drifts.

which is disappearing fast. The thin nickel foil is supported by an acrylic layer. Normally, theacrylic layer is supposed to evaporate before the experiment starts. A possible explanation is, thatthere existed remnants of the acrylic layer, caused by mounting the nickel foil upside-down, withthe acrylic layer facing the target. An arbitrary function is chosen to parameterise the increase inmeasured mean kinetic energy in the mass spectrometer

⟨EL(100, t)

⟩ = ⟨EL0(100)

⟩ + As

(1 − exp

(− ln 2

ts· t

))+ Ac

(1 − exp

(− ln 2

tc· t

)), (2)

where As and Ac describe the increase in mean kinetic energy. The parameters ts and tc corre-spond to half-lives, and 〈EL0(100)〉 is the kinetic energy that would have been measured at thestart of the experiment when the target properties are known.2 The time scale, ts is determinedmainly by the fission rate, causing target heating, which in turn causes diffusion into the Ni-foil,

2 With exception for the remnants of the acrylic layer.

Table 1Fission rates for similar experiments performed at LOHENGRIN. Values of ts for 241Puand 245Cm are extracted graphically from Ref. [15] (see text). The half-life for the in-crease in mean kinetic energy for Cf is estimated to be the same as for Cm

Target material Fission rate Half-life, ts

(1010 fissions/s) (d)

7 µg 245Cm 2 1.8511 µg 241Pu 67 1.1

4 µg 249,250,251Cf 1.3 1.8

and by how much material is lost due to sputtering. In this experiment the fission rate corre-sponds to that reported in Ref. [15]. Therefore, ts was given the same value as for 245Cm(nth, f),see Table 1. For the time constant, tc, describing the influence of the residual acrylic layer weobtained tc = (0.26 ± 0.02)d .

3.2. Electric field drift corrections

The apparent drift in the electric field could only be seen due to the unusual behaviour of theburn-up as a function of time. In order to correct for the electric field drifts the obtained functionsdescribing the changing target, Eqs. (1) and (2), and all the burn-up measurements were used.The relative change in intensity at time t and energy E is then given by

p(EL, t) = √2πσ

I (EL, t)

I (〈t〉) exp

((EL − 〈EL(100, 〈t〉)〉)2

), (3)

where I (EL, t) is the measured number of counts normalised to measurement time and energydispersion. I (〈t〉) is the expected value of the two exponential functions describing the burn-up, at the average time 〈t〉 within its burn-up series, σ is the expected width of the burn-updistributions and 〈EL(100, 〈t〉)〉 is the expected mean energy of the distribution according toEq. (2), see Fig. 4.

To calculate a p-correction value for a non-burn-up energy at a non-burn-up time, linear in-terpolations were performed between the existing p-correction values in time and energy.

3.3. Energy loss correction

Energy loss corrections can only be calculated if the target properties are known. Assumingthat the parameters As , Ac, ts and tc in Eq. (2) are the same for different mass A3 and that allmeasured energies for one mass has the same increase as the mean value, the kinetic energy thatwould have been measured at t = 0, if no increase due to the acrylic layer would exist, is givenby

E0(A) = EL(A, t) − As

(1 − exp

(− ln 2

ts· t

))+ Ac exp

(− ln 2

tc· t

), (4)

where EL(A, t) is the measured kinetic energy.

3 Since the burn-up was measured without parasitical masses, this cannot be checked. However, the same assumptionis made in Ref. [15].

Fig. 4. Example of the calculation of the electric field drift correction values, p(EL, t): the measured value (full squares)is divided by the expected value (open circles) according to Eq. (3), which gives the relative decrease in intensity (fulltriangles) for each measured energy.

The energy loss calculations were performed on the energy E(A) with the computer programSRIM2003 [14]. Two nuclides were considered for each mass, one found when calculating theZucd, assuming uniform charge distribution in the fission-fragments, and the next closest one.The average energy loss value was then used.

3.4. Ionic charge corrections and the Nikolaev–Dimitriev parameters

The correction for the selective measurement at only a few ionic charge states at one kineticenergy has been performed with the semi-empirical formula of Nikolaev and Dimitriev [16],which describes the mean ionic charge state as a function of the nuclear charge Z and the velocityv(Ek,A):

⟨q(v,Z)

⟩ = Z ·[

v′ Z−α

)−1/k]−k

The parameters α, k and v′ in the original work were chosen to: α = 0.45, k = 0.6 and v′ =0.36 cm/ns, and were obtained from experiments with stable isotopes using carbon foils andargon gas. According to Ref. [17], the mean ionic charge state decreases slightly with increasingZ of the target, which is clearly the case here. The experimental values for 〈q(A,E)〉 are obtainedby fitting Gaussians to the q-distributions, which were measured at mean kinetic energy, followedby integration.

With an acrylic layer present, the mean kinetic energy is lower when the fission-fragmentsenter the nickel foil. If the energy loss in the acrylic layer is approximately 6 MeV, the decreasein the mean ionic charge state, according to Eq. (5), would be 0.3, which is indeed the shiftobserved in the data shown in Fig. 5.

Fig. 5. The variation of the mean ionic charge state as a function of mass A. The semi-empirical calculation [16] (shortdashed line) using the original parameters gives to high values. Two other sets of parameters were used, based on theearly and late measurements. Due to internal conversion, some masses (open symbols) show an increase in mean ioniccharge state, an effect, which has been observed for these masses before [15,18].

As a compromise, two sets of parameters for Eq. (5), were used, one for the early measure-ments and another one for the later ones. The parameter v′ was given two different fixed values.4

For some of the masses, one or more isotopes undergo internal conversion, which will change theionic charge state distribution of this mass. Since this occurs after the fission-fragments passedthe nickel foil, their ionic charge state distribution will also have a non-Gaussian contributionand the mean ionic charge state will increase for these masses. This has been reported in previ-ous experiments [15,18]. The masses showing the highest change in mean ionic charge state in245Cm(nth, f) [15], 239Pu(nth, f) [19] and 249Cf(nth, f) [20] also show a similar increase in thiswork. Since these exceptions are not described by Eq. (5), they were omitted in the two fits. Thevalue of the parameters are summarised in Table 2. The results of the fits are shown in Fig. 5.

Once the parameters in Eq. (5) are found, the mean ionic charge can be calculated for otherenergies. This was performed by making a first order Taylor expansion around the measuredenergy, Em,

⟨q(A,E)

⟩ = ⟨q(A,Em)

⟩exp + Z1−α

v′ ·[

v(A,Em)

v′ Z−αucd

)−1/k]−k−1

v(A,Em)

v′ Z−αucd

) 1+kk ·

(1.389

2 · √A · Em

)· (E − Em). (6)

The experimental value determined by fitting the q-distribution is used as the constant term inEq. (6). In this way also the exceptions may be treated in a correct way.

4 There is no convincing reason for choosing the two values. The full analysis was also performed using a different setof parameters, with negligible effect on the final result.

Table 2Nikolaev–Dimitriev parameters obtained from fitting the mean ionic charge state according to Eq. (5)

α k v′ (cm/ns)

Early 0.409 ± 0.002 0.506 ± 0.009 0.5Late 0.572 ± 0.001 0.569 ± 0.004 0.25

3.5. Determination of the emission yield and kinetic energy

The emission yield of a fission-fragment with mass A as a function of kinetic energy and ioniccharge state is given by a double Gaussian:

Y(A,q,E) = Y(A)√2π · σE(A)

· e− (E−〈E(A)〉)22σ2

E(A) · 1√

2π · σq(A)· e− (q−〈q(E,A)〉)2

2σ2q (A) , (7)

where Y(A) is the total yield, σE(A) is the width of the energy distribution, 〈E(A)〉 is the meankinetic energy, σq(A) is the width of the ionic charge state distribution, and 〈q(A,E)〉 is themean ionic charge state. Y(A,q,E) is the experimental value for mass A, ionic charge state q

and energy loss corrected energy E, that has to be corrected according to

Y(A,q,E) = N

�t · EL · p(EL, t) · Cb100(t), (8)

where N is the measured number of events and �t is the measurement time. Since N depends onthe energy interval transmitted at a particle energy EL through the LOHENGRIN spectrometer,which has a constant energy acceptance �EL/EL, the number of events have to be normalised,here divided by EL. p(EL, t) is the correction for the electric field drifts and Cb100(t) the burn-up.

Using Eq. (6), a quantity called q-integrated intensity is defined by re-arranging Eq. (7)

Iq(A,E) = Y(A,q,E) · √2π · σq(A) · e(q−〈q(E,A)〉)2

2σ2q (A)

= Y(A)√2π · σE(A)

· e− (E−〈E(A)〉)22σ2

E(A) . (9)

The q-integrated intensities are calculated for the measured energies in the energy distributions.For each mass this is done by using the σq determined from the fit to its q-distribution. The widthof the q-distribution does not change considerably with energy for a given mass in the energyrange of the fission-fragments [17,21] and varies only little with the atomic number [21]. Themass yields and mean kinetic energies are then found by fitting Gaussians to the q-integratedintensities. The uncertainties given in the q-integrated intensities include both statistical uncer-tainty and systematical uncertainty from the correction functions, where the uncertainty in thep-correction values are the largest. Before fitting the Gaussians an additional 6% uncertainty wasadded linearly because of the systematic uncertainty originating from the burn-up function (fordetails see Appendix C).

3.6. The contribution from 249Cf and the correction to the burn-up curve

3.6.1. Mass yield extractionAs shown in Section 2.1, different Cf isotopes contribute to the detected fission events. How-

ever, if only fission-fragment yields from 249Cf and 251Cf are considered, which always amountto more than 90% (cf. Fig. 2), the count rate of mass A can be written as

Y(A, t) = Y1(A, t) + Y9(A, t), (10)

where Yi(A, t) is the count rate of mass A for each isotope at time t . The subscript i = 1,9stands for 251Cf and 249Cf, respectively. We introduce the total count rate for all isotopes and allfission-fragment masses as the sum Y(t) = ∑124

A=80 Y(A, t), which decreases with time accordingto

Y(t) = CB(t) · Y(0). (11)

Here, we are assuming that there exists a common burn-up function CB(t), this assumption willbe verified later. The contribution of isotope i to the total count rate can be expressed by usingthe fraction fi(t)

5 of its fission rate at time t

Yi(t) = Y(t) · fi(t). (12)

According to the definition above for Y(t), Yi(t) represents the sum over all Yi(A, t). In orderto be able to compare fission-fragment mass distributions from different experiments, one isinterested in the normalised fission-fragment mass distribution yi(A), whose definition is givenby

yi(A) = Yi(A, t)

Yi(t), (13)

with∑

yi(A) = 1 for all light fission-fragments. Combining all equations in this section so far,Eq. (10) can be written as

Y(A, t) = Y(0) · CB(t)(f1(t)y1(A) + f9(t)y9(A)

). (14)

By rearranging, the mass fraction of 251Cf can be found as

y1(A) = Y(A, t)

Y (0) · CB(t)

f1(t)− f9(t)

f1(t)y9(A). (15)

In Eq. (15) the contribution of fission-fragments with mass A resulting from 251Cf is expressedby mostly known or measured quantities. Here, t denotes again the time, when mass A wasmeasured. Since the sum of the left-hand side over all A (in practise from 80 to 124) equals one,Y(0) can be expressed by

Y(0)= 1 + ∑124

A′=80f9(t (A

′))f1(t (A

′)) y9(A′)∑124

A′=80Y(A′,t (A′))

CB(t (A′))·f1(t (A′))

, (16)

where we have replaced t by t (A′) when summing over A′, in order to make clear that for eachmass A′ the time has to be taken, when that particular mass was measured. This concerns the

5 With f1(t) + f9(t) = 1 for all times.

quantities f1(t), f9(t) and CB(t). Inserting Eq. (16) into Eq. (15), and thus eliminating Y(0),leads to

y1(A) = Y(A, t)

CB(t) · f1(t)· 1 + ∑124

A′=80f9(t (A

′))f1(t (A

′)) y9(A′)∑124

A′=80Y(A′,t (A′))

CB(t (A′))·f1(t (A′))

− f9(t)

f1(t)y9(A). (17)

Here the expression Y(A, t)/CB(t) denotes the burn-up corrected count rate. The individualmeasured count rate for a mass A with a certain energy and ionic charge state had to be correctedfor burn-up prior to the determination of the total count rate, due to the rapid burn-up in com-parison to the time of measurement for one mass. For that we used the burn-up correction factorCb100(t) as defined in Eq. (1).

The mass fraction y1(A) for a certain mass A that was measured at time t (A) can be deter-mined now by means of Eq. (17) by using either experimentally obtained data or known data for249Cf. The fission rate fractions f1(t) and f9(t) were calculated as displayed in Fig. 2, but renor-malised to unity. The fission-fragment distributions y9(A) from Refs. [20,22,23] were used aswell as the measured count rates Y(A, t (A)) from this experiment. The burn-up function CB(t)

is a priori not known, and would be equal to Cb100(t), only if the deduced ratio Y1(A, t)/Y9(A, t)

is the same over time, which is definitely not the case. But, as we show below, Cb100(t) can beused to determine CB(t) iteratively.

From the burn-up measurements described in Section 3.1 one knows that the measured countrate for A = 100 as function of time is given by

Y(100, t) = Y(100,0) · Cb100(t). (18)

According to Eq. (14) it is also

Y(100, t) = Y(0) · CB(t) · (f1(t)y1(100) + f9(t)y9(100))

and in particular for t = 0

Y(100,0) = Y(0) · (f1(0)y1(100) + f9(0)y9(100)), (20)

where we have used that CB(0) = 1. Combining Eqs. (18), (19) and (20) leads then to

Cb100(t)= f1(0)y1(100) + f9(0)y9(100)

f1(t)y1(100) + f9(t)y9(100). (21)

Hence, the value of y1(100) obtained with Cb100(t) instead of CB(t) in Eq. (17) allows to calcu-late a better approximation for CB(t). Using that one to calculate a new value for y1(100) leadsto a new CB(t), which in turn generates a new y1(100), etc. This procedure converges fast, i.e.,after 6 iterations the relative change between two successive values of y1(100) is less than 10−6.The ratio between CB(t) and Cb100(t) is displayed in Fig. 6, indicating that the difference is in-creasing with time, reaching about 1.6% after 12 days. Since the measured yields for all masseswere already corrected by division with Cb100(t), the only further correction to be applied isdividing the obtained values for Y(A, t) with the corresponding ratio CB(t)/Cb100(t). Again, t

denotes the mean time for the period when the energy distribution of one mass A was measured.

3.6.2. Mean kinetic energy correctionsMaking the reasonable assumption, that the energy distributions of the fission-fragments from

251Cf and 249Cf have the same width and that their mean values are close, the mean kineticenergy for 251Cf is given by

Fig. 6. Ratio between CB(t) and Cb100(t) as a function of irradiation time, indicating the influence of the isotopicmixture on the burn-up function.

E1(A) = E(A)(y1(A)f1(t) + y9(A)f9(t))

y1(A)f1(t)− E9(A)y9(A)f9(t)

y1(A)f1(t), (22)

where E(A) is the mean kinetic energy determined before isotopic de-convolution.

4. Results

The obtained post-neutron fission-fragment mass distribution of 252Cf∗ at thermal excitation isshown in Fig. 7 together with corresponding data from an evaluation [24] and from spontaneousfission of 252Cf [25]. The mean kinetic energy as a function of post-neutron mass is shown inFig. 8 and the corresponding width of the energy distributions for the Cf-mixture is shown inFig. 9. The data from thermal neutron induced fission are in good agreement with those fromRef. [24], but in the mass range 80 to 98 the measured values are larger than the evaluatedones. The distribution appears to be broader than in the case of spontaneous fission. For a morerealistic comparison the LOHENGRIN data have been 5-point smoothed to mimic the massresolution of the SF data measured with the 2E-technique. The emission yields are enhancedaround A = 115 and diminished around A = 105. This observation confirms findings in Ref. [27],where the reaction 250Cf(t, p) has been used to create 252Cf∗ at a slightly higher average excitationenergy. This is shown in the left part of Fig. 10, where the complementary heavy pre-neutronmasses are shown. Further confirmation comes from a comparison of 252Cf with provisionalmass distributions obtained parasitically from the neutron-induced fission of the decay product251Cf after investigating the reaction 255Fm(nth, f) [28]. The unexpected single-mass peak at A =111 must be attributed to an experimental artefact as carefully being evaluated in the followingchapter. The mean light fragment mass and kinetic energy determined in this work are 〈AL〉 =(107 ± 2) and 〈Ek,L〉 = (103 ± 2) MeV, respectively. The shape of the mean post-neutron kineticenergy distribution as a function of the fragment mass A, follows the expected almost constanttrend, with a steep drop close to symmetric fragment masses [15,19,29–32]. The position of thedrop in mean kinetic energy scales with the fission compound nuclear mass as already reported

Fig. 7. The mass distribution for 251Cf (full squares) compared with an evaluation by England [24], where thegrey-shaded area indicates lower and upper limits. Circles denote results from the spontaneous fission of 252Cf [25].

Fig. 8. The mean kinetic energy for light masses from 249Cf(nth, f) [20,22,23] (open triangles) spontaneous fission of252Cf [25] (open circles) and 251Cf(nth, f) (full squares).

earlier in Ref. [32]. The corresponding data, taken graphically from literature, are shown forillustration in Appendix B.

As expected the kinetic energy for neutron-induced fission is lower than in spontaneous fis-sion. The observed difference between thermal neutron-induced fission and spontaneous fissionof 252Cf around A = 115 might be attributed partly to the much lower mass resolution providedby the 2E-technique, leading to a much smoother energy distribution. However, an increase ofthe kinetic energy close to symmetry has been previously reported [26–28], as it may be seen

Fig. 9. The width of the energy distributions for the Cf-mixture (symbols). The corresponding data from neutron-inducedfission of 250Cf* [20,22,23], 246Cm* [15] as well as from 252Cf(SF) [25] are shown for comparison (see text).

Fig. 10. The mass distribution (left) and total kinetic energy (right) as a function of pre-neutron mass. The data from We-ber et al. [27] are obtained in the reaction 250Cf(t, p) and are compared to the spontaneous fission of 252Cf [25,27]. Thedifference between both spontaneous fission measurements is due to the fact that different values for the normalisationof the total kinetic energy were used.

in the right part of Fig. 10, where excited fission and spontaneous fission of 252Cf is compared,both data obtained with the 2E-technique [27]. The width of the kinetic energy distribution, σE ,is shown in Fig. 9 as a function of the fragment mass A. Also shown are the corresponding datafrom neutron-induced fission of 250Cf* [20,22,23], 246Cm* [15] as well as from 252Cf(SF) [25].The previously observed increase of σE close to symmetric fission is also measured in the presentcase. This is explained as being a superposition of energy distributions from different pre-neutronmasses falling on top of each other after neutron evaporation, which reflects nicely the saw-toothshape of the prompt neutron multiplicity as a function of pre-neutron mass [32,33]. The peakposition indicates a perfect systematic increase when going from 246Cm, A = 118, via 250Cf,

A = 120, to 252Cf, A = 121. The average width, 〈σE〉, is about 2 MeV higher for all masses,whereas the difference between peak and average width is the same. The increase of 〈σE〉 can-not be explained by the superposition of two energy distributions with different mean, since thetabulated mean kinetic energies and widths for the reaction 249Cf(nth, f) cause only a negligibleeffect. Therefore, the observed increase of 〈σE〉 must be attributed to the experimental conditions,as for example target thickness and/or homogeneity.

The final results are summarised in Table 3 in full detail.

5. Discussion

As described in Section 3, several corrections had to be applied to the raw data in orderto derive the fission-fragment mass distribution. All of them are common practise and provenin the data treatment of numerous experiments, except one. The correction for the previouslyunreported, but apparent electric field drift of the mass spectrometer needs some additional com-ments. Is it possible to apply corrections using the p-correction values if A �= 100 and q �= 22?Since the electric field is proportional to the kinetic energy, it is reasonable to assume that thecorrections may be applied also for another mass at q = 22. No time dependence on the meanionic charge states was found. Hence, the corrections were also applied for q �= 22. Unfortu-nately, since the burn-up was measured with field settings not allowing the analysis of anyparasitical mass, this assumption could not be proven directly. Hence, the data analysis wasrepeated without applying the p-correction and is compared with the final results in Fig. 11.The masses in the range from 98 to 104 and 116 to 118 show statistically significant differ-ences.

A second way to check the quality of all corrections applied to the raw data, in particularthe correction to the electric field drift, is offered by the LOHENGRIN mass spectrometer itself.The separation A/q and E/q allows in certain cases so-called “parasitic” particles with thesame ratios, A/q , to be detected. They are measured at a different time than when measureddirectly, and hence, the electric field drift correction values are different. The other correctionfunction values are obviously also different. We calculated q-integrated intensities according toEq. (9) for the parasitic masses with |q − 〈q〉| < 2. This was also done without any electricfield correction values for comparison. When plotting these values together with the q-integratedintensities of the direct measurement, the necessity of doing the electric field drift correctionsbecomes evident, although in some cases no significant improvement was seen. As an exampleenergy distributions for mass 100 are shown with and without electric field drift corrections inFig. 12.

Fig. 11 exhibits that the exceeding yield at mass A = 111 is correlated with a relatively lowmean kinetic energy compared to neighbouring masses. The uncertainties on the data points areestimated taking statistical and all known and relevant systematical uncertainties into accountusing the propagation-of-error formula. Details are given in Appendix C. Due to the appliedadditional field-drift corrections the error bars appear already larger than for other reported LO-HENGRIN data, but cannot explain the data for A = 111 as a statistical effect. Since we haveno explanation for the apparent deviation, we must not exclude the data point as we obtainedit by applying exactly the same analysis procedure. However, the deviation is very likely anexperimental artefact.

Although the TKE of the fission-fragments was not measured, it can be estimated using themeasured post-neutron mean kinetic energy and post-neutron mean light mass. The pre-neutron

Table 3Light fission-fragment yields, mean kinetic energies and widths from the thermal neutron-induced fission of 251Cf. Thewidths are obtained from the fits to the q-distribution and correspond to the mixture of isotopes present in this experiment.The uncertainty before correcting for the influence of 249Cf is also shown, labelled as ‘with 249Cf’

Mass Yield Uncertainty Total 〈E〉 Uncertainty Total σE

% with 249Cf (MeV) with 249Cf (MeV)

80 0.028 0.002 0.002 106.3 0.6 0.7 8.9 ± 0.681 0.051 0.003 0.004 106.5 0.7 0.8 9.2 ± 0.682 0.076 0.005 0.005 105.7 0.6 0.7 9.0 ± 0.583 0.147 0.008 0.01 104.7 0.6 0.7 9.6 ± 0.784 0.17 0.01 0.02 106.1 0.5 0.6 9.1 ± 0.585 0.29 0.02 0.02 104.8 0.5 0.6 8.8 ± 0.586 0.34 0.02 0.03 105.2 0.5 0.6 9.0 ± 0.587 0.51 0.03 0.04 103.8 0.5 0.6 8.8 ± 0.588 0.51 0.03 0.03 104.8 0.5 0.6 8.4 ± 0.489 0.61 0.03 0.04 105.1 0.4 0.5 8.4 ± 0.490 0.75 0.04 0.05 104.9 0.4 0.5 8.0 ± 0.491 0.87 0.05 0.06 105.0 0.4 0.5 7.9 ± 0.492 1.00 0.05 0.07 104.8 0.4 0.5 8.0 ± 0.493 1.36 0.07 0.09 104.8 0.4 0.5 7.9 ± 0.494 1.6 0.1 0.1 104.2 0.4 0.6 8.2 ± 0.495 1.9 0.1 0.2 103.8 0.5 0.6 8.3 ± 0.496 2.0 0.1 0.2 104.2 0.5 0.6 8.2 ± 0.497 2.3 0.2 0.2 103.7 0.5 0.6 8.2 ± 0.498 2.1 0.2 0.2 103.6 0.4 0.6 7.7 ± 0.399 2.2 0.2 0.2 104.5 0.3 0.4 7.5 ± 0.3

100 3.0 0.2 0.2 104.1 0.3 0.4 7.9 ± 0.3101 3.5 0.2 0.2 102.9 0.4 0.4 7.9 ± 0.3102 3.4 0.2 0.3 103.1 0.4 0.6 7.8 ± 0.3103 4.3 0.3 0.3 103.2 0.4 0.5 7.5 ± 0.3104 4.6 0.3 0.4 102.6 0.4 0.6 7.7 ± 0.3105 4.6 0.3 0.4 102.6 0.5 0.7 7.9 ± 0.4106 4.7 0.3 0.4 102.9 0.5 0.7 8.0 ± 0.4107 4.0 0.3 0.4 102.4 0.5 0.7 7.7 ± 0.3108 4.4 0.3 0.3 102.2 0.4 0.5 8.0 ± 0.3109 4.7 0.3 0.4 102.0 0.4 0.5 8.1 ± 0.3110 4.9 0.3 0.4 102.2 0.4 0.5 8.0 ± 0.3111 6.0 0.3 0.4 100.6 0.4 0.5 8.2 ± 0.3112 4.6 0.3 0.3 101.6 0.4 0.5 8.7 ± 0.4113 4.4 0.3 0.3 102.1 0.4 0.5 8.6 ± 0.3114 4.0 0.2 0.3 102.0 0.4 0.5 8.4 ± 0.3115 3.7 0.2 0.3 102.6 0.4 0.5 8.4 ± 0.3116 3.9 0.2 0.3 101.7 0.5 0.5 9.1 ± 0.4117 2.5 0.2 0.2 101.2 0.4 0.5 8.9 ± 0.3118 2.0 0.1 0.2 101.3 0.5 0.6 9.5 ± 0.4119 1.5 0.1 0.1 102.2 0.5 0.6 9.4 ± 0.4120 1.0 0.1 0.1 101.6 0.5 0.6 10.0 ± 0.5121 0.61 0.03 0.04 100.4 0.7 0.8 11.3 ± 0.6122 0.43 0.04 0.05 95 2 2 13 ± 2123 0.27 0.02 0.03 94 2 2 11 ± 1124 0.29 0.04 0.05 94 2 3 10 ± 2

Fig. 11. Mass distribution and mean kinetic energies with (full squares) and without (open circles) applying electric fielddrift corrections.

Fig. 12. The q-integrated intensity as a function of kinetic energy for A = 100, with (left) and without (right) electricfield drift corrections. The result of both direct and parasitic measurements are shown.

mean light mass is also needed and must be taken from another experiment. The average TKE isapproximately given by

〈TKE〉 = ApreL

ApostL

· EpostL · Acn

Acn − ApreL

, (23)

where ApreL is the pre-neutron mean light mass, A

postL is the post-neutron mean light mass

and Acn is the compound nuclear mass. Using the light pre-neutron mass from the reaction250Cf(t, p) [27], AL = 110.2, gives 〈TKE〉 = (188.5 ± 5.1) MeV and using the light pre-neutronmass from spontaneous fission [33], AL = 108.6, gives 〈TKE〉 = (183.7 ± 5.0) MeV. Thesevalues agree well with the mean TKE obtained by using the formula given by Viola [34],(188.1 ± 2.2) MeV and the experimental value from the reaction 250Cf(t, p) in Ref. [27], 189.1MeV (no uncertainty given).

Different fissioning systems show a systematic increase of the post-neutron average light masswhile the post-neutron average heavy mass stays constant at AH = (139 ± 1) [35]. The system-atics gives an estimated post-neutron mean light mass of AL = (109 ± 1) and agrees well withour value of (107 ± 2).

The mean kinetic energy for masses close to symmetry (corresponding to the heavy massAH = 132) is significantly higher for the neutron-induced fission compared to spontaneousfission. A similar sharp decrease close to symmetry is also seen in the reaction 249Cf(nth, f)[20,22,23], see Fig. 8.

5.1. Interpretation of differences in terms of fission mode weights

A fission mode in the multi-modal random neck-rupture model [4] is a local minimum in thenuclear landscape connecting saddle and scission point. Each mode has a characteristic TKEand mean mass. In the frame of the multi-modal random neck-rupture model, mass and energydistributions have been described for the actinide region as a function of incident neutron energywith in principal two asymmetric fission modes [5–8]. One fission mode, SI, which has a morecompact shape before scission and thus a higher TKE and, a more deformed fission mode, SII,with lower TKE. For spontaneous fission of 252Cf, additional modes, mainly a “super-deformed”SX mode, had to be added in order to reasonably describe the experimental data [9].

The interpretation of the fission-fragment properties in terms of fission modes is only possibleif the full pre-neutron mass distribution and corresponding kinetic energies are known. However,neutron evaporation is in the order of a few neutrons, which means that changes in the fissionmode weights between the spontaneous fission of 252Cf and the fission of 252Cf∗ at thermalexcitation might still be observed.

According to Ref. [36] the mean pre-neutron mass and TKE for the fission modes are:SI 116.4 amu/194.5 MeV, SII 109.0 amu/186.7 MeV, SX 105.1 amu/177.0 MeV, SIII94.5 amu/157.9 MeV and SL 126 amu/184.8 MeV. Looking at the post-neutron mass distri-bution in Fig. 7 the yields are enhanced around A = 115 and A = 90 and diminished aroundA = 105. The mean kinetic energy, see Fig. 8, is enhanced in the region around A = 115, but nodifference is seen elsewhere. Mass yield and kinetic energy increase around A = 115 stronglyindicates, that for neutron induced fission, the SI fission mode is enhanced and the SII/SX fissionmodes are suppressed. The SII fission mode is suppressed also in comparison to the SX fissionmode, because the yields are enhanced around A = 90, while the kinetic energy does not change.If SIII would be enhanced to explain the increase in yield around A = 90, the kinetic energywould also be lower around A = 90, which is not the case. In other words, our experimentalresults support the existence of an additional SX fission mode in 252Cf.

6. Conclusion

For the first time the light fission-fragment post-neutron mass distribution from the reaction251Cf(nth, f) was measured with high mass-resolution. The determined mean light fragment mass,〈AL〉 = (107 ± 2) and the mean light kinetic energy 〈Ek,L〉 = (103 ± 2) MeV agree well withexpectations. The contribution from the reaction 249Cf(nth, f) was properly taken into accountand corrected for. A comparison with the spontaneous fission of 252Cf revealed an increase inyield above mass number A = 113 and a decrease around A = 105. The average kinetic en-ergy is higher for A > 114. As expected the kinetic energy of the light fragments is higherfor spontaneous fission than for thermal-neutron induced fission. The experimental results arein good agreement with an existing evaluation [24] and experimental data from the reaction250Cf(t, p) [27]. The increased width of the kinetic energy distribution around A = 121 confirmsresults observed in other neutron-induced fission reactions. A comparison with neighbouringisotopes show, that the peak position correlates well with the mass of the fissioning nucleus.

The observed changes in the fission-fragment distributions may be well understood withinthe multi-modal random neck-rupture model [4]. Within this model description, there is a stronghint for an increased contribution of a super-deformed fission mode in thermal-neutron inducedfission of 252Cf*, giving support to the existence of an additional SX fission mode in 252Cf.

Since the available sample material does not allow a 2E measurement to obtain pre-neutronfission-fragment distributions, with, e.g., an ionisation chamber, a detailed Monte Carlo simula-tion will have to be performed to get a more quantitative interpretation in terms of fission modes.

Acknowledgements

We are indebted to N. Trautmann and co-workers from the Institute for Nuclear Chemistry ofthe Mainz University for the target preparation. The target material used in this work was suppliedby the Product Development and Cf Program Group, Nuclear Science and Technology Division,Oak Ridge National Laboratory (ORNL) under the Transplutonium Element Production (TEP)Program funded by the US Department of Energy Office of Science (DOE-SC). ORNL is oper-ated by UT-Battelle, LLC for the USDOE under contract No. DE-AC05-00OR22725. The majorpart of the analysis was funded by the European Commission by granting a fellowship for one ofthe authors (E.B.).

Appendix A. The mass distribution from the reaction 249Cf(nth, f)

The mass yields from the reaction 249Cf(nth, f) were used in the analysis to extract the yieldand kinetic energy from the reaction 251Cf(nth, f). It is shown together with an evaluation byEngland [24] in Fig. A.1. The experimental yield for masses in the range from 89 to 95 also seemsto be slightly higher than the corresponding value from the evaluation [24]. The kinetic energyfrom the reaction 249Cf(nth, f) is shown in Fig. 8. The drop in kinetic energy near symmetry isalso seen in this reaction.

Appendix B. Mean kinetic energy distribution, 〈E(A)〉, for 233U(nth, f) and 235U(nth, f)

The mean fragment kinetic energy as a function of the fragment mass A for the reactions233U(nth, f) and 235U(nth, f) [32] is shown in Fig. B.1. A comparison of the position of the steepdrop in mean kinetic energy, 〈E(A)〉 shows, that it scales with the compound nuclear mass. Thisis exactly what was found for the reactions 249Cf(nth, f) and 251Cf(nth, f).

Appendix C. Uncertainty estimations

The program Origin [37] was used to fit all functions to the experimental data. Origin usesa least square fit routine based on the Levenberg–Marquardt algorithm. The program providesuncertainties on the parameters found by the fit, as well as correlations between the parame-ters. The correlation was in all cases found to be small and was in no case taken into account.The contribution of the different parameters when correcting the data was calculated using thepropagation-of-error (PoE) formula.

Although the electric field settings are assumed to be correct after formation some deviationsseems to still exist. In Fig. C.1 the selected burn-up values are shown relative to the fit. The un-certainty region of the function describing the burn-up is shown in grey. The experimental valueshere have only the uncertainty from the Gaussian fit, which is in the same order as the statistical

Fig. A.1. The light fission-fragment mass distribution for 249Cf [20,22,23] is compared to an evaluation by England [24].The data from Djebara [20] have been renormalised with a factor of 1.022, since Djebara did not measure the entire massdistribution.

Fig. B.1. Comparison of the mean fragment kinetic energy as a function of the fragment mass A for the reactions233U(nth, f) and 235U(nth, f) [32]. The data have been taken graphically from the figures presented in Ref. [32].

uncertainty. The observed spread of the individual burn-up data has a standard deviation of 6%and was taken into account by increasing the uncertainty of the q-integrated values before fittingthe final distributions.

The uncertainty of the p-correction values in Section 3.2 consists of a statistical and theuncertainty of the parameters of those functions describing the burn-up and the increase of themeasured kinetic energy, see Fig. 3. In addition comes the uncertainty of the linear interpolations

Fig. C.1. The selected burn-up measurements divided by the value from the function describing the burn-up show morespread then expected. An extra 6% uncertainty was added linearly to the q-integrated values.

performed between the p-corrections values. All these contributions were considered using thePoE formula.

The uncertainty of the mean ionic charge, Eq. (6), was taken into account by giving the secondterm, the deviation from the experimental value, an uncertainty of 10% of its value. This was thenadded quadratically to the experimental uncertainty.

The total yield and mean kinetic energy is the result of a Gaussian fit to the q-integrated distri-bution. The uncertainty of the q-integrated values has a statistical part and a part that originatesfrom all correction functions. There exists an obvious correlation between the p-correction val-ues and the burn-up correction values, but they were treated as if they were uncorrelated, sincethe particular correlation is unknown.

The uncertainty introduced when calculating the influence of 249Cf was determined using thePoE formula. This is the final uncertainty quoted in Table 3. The uncertainty before the extractionis also quoted.

The calculation of the influence of 249Cf was also performed by maximising and minimis-ing the influence of 249Cf, using the upper and lower limits of the cross-sections [11,12]. Thedifference was negligible and the contribution to the uncertainty from the cross-sections of theCf-isotopes was therefore not taken into account.

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Light fission-fragment mass distribution from the reaction 251Cf(nth, f )

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