Measurements of $^{233}$U$(\text {n}_{\text {th}},\text {f})$ fission product mass yields with the LOHENGRIN recoil mass spectrometer

Abstract

Numerous measurements of fission product yields were performed since the discovery of the nuclear fission process. However, more precise and reliable fission product yields are requested. Lack of covariance matrices make difficult to use it for specific application purposes such as the propagation uncertainty of decay heat. In this work, we propose to measure independently the fission product mass yields for the whole heavy peak (including the symmetric mass region) for the $^{233}$U$(\text {n}_{\text {th}},\text {f})$ reaction. Both average values and experimental covariance is provided. The fission product mass yields are measured with the LOHENGRIN recoil mass spectrometer of the ILL using an ionization chamber located at the focal plane. A new procedure of data taking has been developed in order to minimize the biases. Concretely several ionic charges and kinetic energy distributions have been measured for each mass. Particular attention has been considered in the monitoring of the target time evolution. Additional corrections were necessary in the symmetry mass region due to contaminants coming from the LOHENGRIN recoil mass spectrometer. A complex Monte Carlo analysis has been developed in order to better propagate all the uncertainties. The fission product mass yields of the $^{233}$U$(\text {n}_{\text {th}},\text {f})$ and its associated covariance matrix has been produced. An overall good agreement has been observed with ENDF/B-VIII.0 in contrast with the JEFF-3.3 evaluation. A precision around 2% for the heavy peak has been measured. The experimental covariance matrix was also computed. In the symmetry mass region, two components were observed in the kinetic energy distribution. One of this component was considered as an artifact and was ruled-out.

Appendices

Appendix A: Glossary

This section summarizes all the notations used in the analysis part of the article.

A::

mass of the nucleus $^{A}_{Z}$X

Z::

nuclear charge of the nucleus $^{A}_{Z}$X

q::

ionic charge

$q_{i}$ or $q^{\times }$::

ionic charge for which kinetic energy distributions are performed

$E_{k}$::

kinetic energy

$\varDelta E_{k}$::

kinetic energy resolution

$E_{k}^{\times }$::

kinetic energy for which ionic charge distributions are performed

$\overline{E_{k}}$::

mean kinetic energy extracted from the kinetic energy distribution

$\sigma _{E_{k}}$::

standard deviation extracted from the kinetic energy distribution

t::

time when the measurement is made since the beginning of the experimental campaign

$\varDelta t_{m}$::

measuring time

$BU(t)$ (Burn-up)::

constructed observable to follow the loss of fissile material from the target as a function of time

$N(A,q,E_{k},\varDelta t_{m},t)$::

number of counts extracted from the ionization chamber

$\mathcal {N}\left( A,q_{i}\right)$::

relative mass yield calculated from the kinetic energy distribution measured with the ionic charge $q_{i}$

P(q)::

ionic charge probability derived from the ionic charge distribution

$\text {Cov}\left( \mathcal {N}\left( A,q_{i}\right) ,\mathcal {N}\left( A,q_{j}\right) \right)$::

element of the covariance matrix between the relative mass yield calculated from the kinetic energy distributions measured at $q_{i}$ and $q_{j}$

$f(E_{k},\mu ,\sigma ,\alpha )$::

Gaussian law with an exponential tail

$\overline{\mathcal {N}}\left( A\right)$::

average relative mass yield

$\overline{\mathcal {N}_{0}}\left( A\right)$::

reference average relative mass yield

r::

cross normalized factor

$\overline{\mathcal {N}^{\times }_{1}}\left( A\right)$::

cross normalized average relative mass yield in respect to reference data set

$\tilde{\mathcal {N}}\left( A\right)$::

average relative mass yield after combining cross normalized average mass yield and the reference average mass yield

Y(A)::

absolute mass yield

$A_{cont}$::

contaminant mass

$q^{pre}_{cont}$::

contaminant ionic charge before the charge exchange

$P_{Im}$::

probability to detect the contaminant inside the ionization chamber

$P_{q\rightarrow q^{'}}$::

charge exchange probability

$\beta _{i=1,2}$::

free parameters used in the ”extremum method”

$N\left( A,q,E_{k},\varDelta t_{m},t\right) _{meas}$::

number of counts extracted from the ionization chamber which include the contaminant and the symmetric mass signal

Appendix B: Determination of charge changing probabilities in the symmetry mass region

The charge changing probabilities were measured using two High Purity Germanium detectors surrounded an ionization chamber in the experimental area 2. To do so, we used masses with strongly populated microsecond isomers that are clearly identified by ion-delayed gamma ray coincidences. Since the ions arrive in their excited microsecond state in the focal plane, we can isolate the charge exchange by rest-gas collisions alone without perturbation from nuclear decays. The idea is to mimic the charge exchange in the LOHENGRIN spectrometer. To do so the main magnet was set to select the ratio $A/q_{0}$. Then, different settings were applied for the condenser to select the ratio $E_{k}/q$ with $q=q_{0}\pm 1$ and $q=q_{0}\pm 2$. In this way, we mimic the loss or capture of one or two electrons between the main magnet and the condenser. Table 2 shows the results for two heavy isotopes $^{136}$Xe and $^{132}$Te. These isotopes were identified by using a time coincidence between the IC and the HPGe. Indeed both isotopes have microsecond states, which are easy to extract without any background issue. However, this measurement is performed for one ionic charge and few isotopes at a given time (for a given gas pressure).

Table 2 Charge exchange probability measured with the LOHENGRIN spectrometer

Fig. 19

Analysis process. Same data are used multiple times. Propagation of uncertainties is then more complex. A dedicated Monte Carlo code is developed

Fig. 20

Time evolution of the burn-up (top) and the kinetic energy parameters (bottom). The red points refer to the experimental data, whereas the blue line refers to fitting functions. The asymptotic values for the three kinetic energy parameters are also mentioned

Appendix C: Details on Monte Carlo data analysis

Propagation of uncertainties through Eqs. (5), (8) and (11) is difficult to assess analytically. Figure 19 summarizes the analysis process. A solution is to develop a Monte Carlo code to perform this propagation. Determination of the target burn-up is firstly presented. Then the dedicated Monte Carlo code is detailed for the heavy or light peak region (yield higher than 1.5%) and the symmetry mass region. This analysis is used for the ”Symmetry” campaign performed in September 2014. Analysis for other campaigns is detailed in [26, 27].

1.1 Appendix C.1: Burn-up analysis

For each target, the BU evolves differently with time. Figure. 20 (top) shows the BU time evolution. It is normalized to the first measurement. In this case, the BU is fitted by:

$$\begin{aligned} BU(t)=e^{p_{0}t+p_{1}}+p_{2}t+p_{3} \end{aligned}$$

(C.1)

The time evolution of BU parameters $BU_{\mu }, \ BU_{\sigma }, \ BU_{\alpha }$ was also fitted using a combination of polynomial and exponential functions (see the bottom part of Fig. 20). The asymptotic values are also indicated in the figure. The fitting parameters are then sampled and an estimation of the burn-up is then performed for each sampling. Note that a systematic uncertainty must be added for each point to not reject the fit. Figure 21 presents the evolution of the p-value, which indicates the goodness of the fit, as a function of the systematic uncertainty. In this case, we choose a significance level of $99.7\%$ which corresponds to an absolute systematic uncertainty of $\varDelta _{syst}=0.008$. The relative systematic uncertainty is between $1\% \le \varDelta _{syst} \le 3\%$, according to the experimental points. This additional uncertainty can be interpreted as the LOHENGRIN stability and reproducibility. Once the fit is not rejected, parameters $\left\{ p_{i}\right\}$ and related covariance matrix Cov can be extracted. To be used in the Monte Carlo code analysis, the parameters must be independent. Decorrelation is done by applying the following equations:

$$\begin{aligned} \theta =Cov^{-1/2}P=RD^{-1/2}R^{-1}P \end{aligned}$$

(C.2)

with $\theta$ the vector of free parameters $\left\{ z_{i}\right\}$, P the vector of correlated parameters $\left\{ p_{i}\right\}$, D a diagonal matrix with eigenvalues of Cov and the column vectors of R the related eigenvectors. It can be shown that the covariance matrix of the free parameters is the identity matrix $I_{n}$.

Fig. 21

BU of the ”Symmetry” campaign performed in September 2014 with a $^{233}$U target (left). Evolution of the p-value as a function of additional systematic uncertainty (right). For a confidence level of, $99.7\%$ the systematic uncertainty is equal to 0.006 for each experimental point

The same procedure is performed for the relative mean kinetic energy evolution $\varDelta BU_{E}(t)$ and the relative width evolution $\varDelta BU_{\sigma _{E}}(t)$.

1.2 Appendix C.2: Mass yield analysis: regular case

The idea is to do a meta-analysis of the experiment. This is performed by sampling the free parameters until convergence of mean value and variance are obtained. The principle is the following: since the number of counts measured with the IC obeys statistical laws, we sample the random variable $N\left( A,q, E_{k},\varDelta t_{m}\right)$ through a Poisson distribution with a mean value corresponding to the IC count rate. We performed it for each measured point of all distributions. Concerning the BU, first the free parameters $\theta$ are sampled following a Gaussian distribution $\mathcal {N}\left( \overline{\theta },1\right)$. Then, the inverse of (C.2) is calculated:

$$\begin{aligned} P=Cov^{1/2}Z \end{aligned}$$

(C.3)

For each sampling, we apply all equations from Eqs. (5) to (9). Then, we perform an average over all the samples. Finally, the cross normalization with Martin et al. data [26] is achieved. All the details are shown in the following subsection. Converged mass yields and uncertainties are obtained after $10^{5}$ samplings of each experimental point.

1.2.1 Appendix C.2.1: Relative mass yield

After sampling the number of counts $N(A,q, E_{k})$ through a Poisson distribution and decorrelated the BU parameters, we calculate the correlation coefficients using Eq. (5). Then, the relative mass yield $\mathcal {N}_{k}\left( A,q_{i}\right)$ is calculated according to (5). For the averaging step, we need to calculate the covariance matrix $\varvec{C}$ between $\mathcal {N}\left( A,q_{i}\right)$:

$$\begin{aligned} C_{ij}= & {} Cov\left( \mathcal {N}\left( A,q_{i}\right) ,\mathcal {N}\left( A,q_{j}\right) \right) \nonumber \\= & {} \frac{1}{n-1}\sum _{k=0}^{n}\left( \mathcal {N}\left( A,q_{i}\right) -\mathcal {N}_{k}\left( A,q_{i}\right) \right) \nonumber \\&\times \left( \mathcal {N}\left( A,q_{j}\right) -\mathcal {N}_{k}\left( A,q_{j}\right) \right) \end{aligned}$$

(C.4)

with

$$\begin{aligned} \mathcal {N}\left( A,q_{i}\right) =\frac{1}{n}\sum _{k=0}^{n}\mathcal {N}_{k}\left( A,q_{i}\right) \end{aligned}$$

(C.5)

and n the number of samples. The average relative mass yield for each sampling $\overline{\mathcal {N}}_{k}(A)$ is:

$$\begin{aligned} \overline{\mathcal {N}}_{k}(A)=\left( \sum _{i,j}\varvec{C^{-1}_{ij}}\right) ^{-1}\left( \sum _{i,j}C^{-1}_{ij}\mathcal {N}_{k}\left( A,q_{j}\right) \right) , \end{aligned}$$

(C.6)

and the average relative mass yield is:

$$\begin{aligned} \overline{\mathcal {N}}(A)=\frac{1}{n}\sum _{k=0}^{n}\overline{\mathcal {N}}_{k}(A) \end{aligned}$$

(C.7)

Note that if the $\chi ^{2}$ test fails, the additional uncertainty $\varDelta _{\text {add}}$ is taken into account through the count rate uncertainty. In this case, the count rate doesn’t follow a Poisson law anymore, rather a truncated positive Gaussian distribution with:

$$\begin{aligned} \mu= & {} N(A,q,E_{k},\varDelta t_{m},t) \nonumber \\ \sigma ^{2}= & {} N(A,q,E_{k},\varDelta t_{m},t)+\varDelta _{\text {add}}^{2} \end{aligned}$$

(C.8)

Finally, the covariance between average relative mass yield is:

$$\begin{aligned}&Cov\left( \overline{\mathcal {N}}\left( A_{i}\right) ,\overline{\mathcal {N}}\left( A_{j}\right) \right) = \frac{1}{n-1} \nonumber \\&\quad \times \sum _{k=0}^{n}\left( \overline{\mathcal {N}}\left( A_{i}\right) -\overline{\mathcal {N}_{k}}\left( A_{i}\right) \right) \left( \overline{\mathcal {N}}\left( A_{j}\right) -\overline{\mathcal {N}_{k}}\left( A_{j}\right) \right) \end{aligned}$$

(C.9)

1.2.2 Appendix C.2.2: Cross normalization

In this section, we will detail the way to calculate the covariance matrix after the cross normalization. We remind that the average relative mass yield after the cross normalization is written:

$$ \begin{aligned}&\text {if} \ A \in M_{0} \ \& \ A \notin M_{1}, \ \tilde{\mathcal {N}}(A) = \overline{\mathcal {N}_{0}}(A) \nonumber \\&\text {if} \ A \in M_{1} \ \& \ A \notin M_{0}, \ \tilde{\mathcal {N}}(A) = \overline{\mathcal {N}^{\times }_{1}}(A) = r\times \overline{\mathcal {N}_{1}}(A) \nonumber \\&\quad \text {with,} \nonumber \\&r = \left( \varvec{\overline{\mathcal {N}^{\times }}}_{c_{1}}^{t}\varvec{\varOmega }^{-1}\varvec{\overline{\mathcal {N}^{\times }}}_{c_{1}}\right) ^{-1}\varvec{\overline{\mathcal {N}^{\times }}}_{c_{1}}^{t}\varvec{\varOmega }^{-1}\varvec{\overline{\mathcal {N}}}_{c_{0}} \nonumber \\&\text {if} \ A \in M_{0} \cup M_{1}, \ \tilde{\mathcal {N}}(A) = <\overline{\mathcal {N}_{0}}(A) |h|\overline{\mathcal {N}^{\times }_{1}}(A)> \end{aligned}$$

(C.10)

Therefore, the covariance matrix is the juxtaposition of 6 submatrices. Since each data set is independent, the first submatrix is:

$$ \begin{aligned}&\text {if} \ A_{i} \in M_{0} \ \& \ A_{i} \notin M_{1}, \ \text {and} \ A_{j} \in M_{1} \ \& \ A_{j} \notin M_{0} \nonumber \\&\quad Cov\left( \tilde{\mathcal {N}}(A_{i}),\tilde{\mathcal {N}}(A_{j})\right) = 0 \end{aligned}$$

(C.11)

The second submatrix is:

$$ \begin{aligned}&\text {if} \ A_{i} \in M_{0} \ \& \ A_{i} \notin M_{1}, \ \text {and} \ A_{j} \in M_{0} \ \& \ A_{j} \notin M_{1} \nonumber \\&\quad Cov\left( \tilde{\mathcal {N}}(A_{i}),\tilde{\mathcal {N}}(A_{j})\right) = Cov\left( \overline{\mathcal {N}}(A_{i}),\overline{\mathcal {N}}(A_{j})\right) \nonumber \\ \end{aligned}$$

(C.12)

Before going further, it is necessary to calculate the correlation between r and the relative common mass of both data set. Note that since the way to find r is iterative, here we will consider only the first step for the covariance calculation. We can rewrite r:

$$\begin{aligned} r= & {} \frac{r_{0}}{r_{1}} \nonumber \\ r_{0}= & {} \varvec{\overline{\mathcal {N}}}_{c_{1}}^{t}\varvec{\varOmega }^{-1}\varvec{\overline{\mathcal {N}}}_{c_{0}} \nonumber \\ r_{1}= & {} \left( \varvec{\overline{\mathcal {N}}}_{c_{1}}^{t}\varvec{\varOmega }^{-1}\varvec{\overline{\mathcal {N}}}_{c_{1}}\right) \end{aligned}$$

(C.13)

Then,

$$\begin{aligned} \frac{\partial r}{\partial \overline{\mathcal {N}}_{1}(A_{i})}= & {} \frac{1}{r_{1}}\delta _{i}\varvec{I}\varvec{\varOmega }^{-1}\varvec{\overline{\mathcal {N}_{c_{0}}}} \nonumber \\&-\frac{r}{r_{1}}\left( \delta _{i}\varvec{I}\varvec{\varOmega }^{-1}\varvec{\overline{\mathcal {N}_{c_{1}}}}+\varvec{\overline{\mathcal {N}_{c_{1}}}^{t}}\varvec{\varOmega }^{-1}\delta _{i}\varvec{I}\right) \nonumber \\ \frac{\partial r}{\partial \overline{\mathcal {N}}_{0}(A_{i})}= & {} \frac{1}{r_{1}}\varvec{\overline{\mathcal {N}_{c_{1}}}^{t}}\varvec{\varOmega }^{-1}\delta _{i}\varvec{I} \end{aligned}$$

(C.14)

with $\varvec{I}$ the unity vector. The third submatrix is written:

$$ \begin{aligned}&\text {if} \ A_{i} \in M_{1} \ \& \ A_{i} \notin M_{0}, \ \text {and} \ A_{j} \in M_{1} \ \& \ A_{j} \notin M_{0} \nonumber \\&\quad Cov\left( \tilde{\mathcal {N}}(A_{i}),\tilde{\mathcal {N}}(A_{j})\right) =\overline{\mathcal {N}}(A_{i})\overline{\mathcal {N}}(A_{j})Cov(r,r) \nonumber \\&\qquad \times r\overline{\mathcal {N}}(A_{j})Cov(\overline{\mathcal {N}}(A_{i}),r) \nonumber \\&\qquad \times r\overline{\mathcal {N}}(A_{i})Cov(r,\overline{\mathcal {N}}(A_{j})) \nonumber \\&\qquad \times r^{2}Cov\left( \overline{\mathcal {N}}(A_{i}),\overline{\mathcal {N}}(A_{j})\right) \end{aligned}$$

(C.15)

with,

$$\begin{aligned} Cov(r,r)= & {} \sum _{k}\sum _{l}\frac{\partial r}{\partial \overline{\mathcal {N}}_{1}(A_{k})}\frac{\partial r}{\partial \overline{\mathcal {N}}_{1}(A_{l})}\nonumber \\&\times Cov(\overline{\mathcal {N}}_{1}(A_{k}),\overline{\mathcal {N}}_{1}(A_{l})) \nonumber \\&+ \sum _{k}\sum _{l}\frac{\partial r}{\partial \overline{\mathcal {N}}_{0}(A_{k})}\frac{\partial r}{\partial \overline{\mathcal {N}}_{0}(A_{l})}\nonumber \\&\times Cov(\overline{\mathcal {N}}_{0}(A_{k}),\overline{\mathcal {N}}_{0}(A_{l})) \end{aligned}$$

(C.16)

because each data set is independent. Also,

$$\begin{aligned} Cov\left( r,\overline{\mathcal {N}}_{1}(A_{j})\right) = \sum _{l}\frac{\partial r}{\partial \overline{\mathcal {N}}_{1}(A_{l})}Cov\left( \overline{\mathcal {N}}_{1}(A_{l}),\overline{\mathcal {N}}_{1}(A_{j})\right) \nonumber \\ \end{aligned}$$

(C.17)

The weighted average between two different data sets can be written:

$$\begin{aligned} \tilde{\mathcal {N}}(A_{i})= & {} \left[ Cov^{-1}(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}))\right. \nonumber \\&+ 2\times Cov^{-1}(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{i})) \nonumber \\&+ \left. Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{i}))\right] ^{-1} \nonumber \\&\times \left[ \overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})Cov^{-1}(\overline{\mathcal {N^{\times }}}_{c_{1}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}))\right. \nonumber \\&+ \overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})) \nonumber \\&+ \overline{\mathcal {N}}_{c_{0}}(A_{i})Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})) \nonumber \\&+ \left. \overline{\mathcal {N}}_{c_{0}}(A_{i})Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{i}))\right] \end{aligned}$$

(C.18)

with,

$$\begin{aligned} Cov(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{j})) = \overline{\mathcal {N}}_{c_{1}}(A_{i})Cov(r,\overline{\mathcal {N}}_{c_{0}}(A_{j}))\nonumber \\ \end{aligned}$$

(C.19)

and,

$$\begin{aligned} Cov(r,\overline{\mathcal {N}}_{c_{0}}(A_{j})) \!=\! \sum _{l}\frac{\partial r}{\overline{\mathcal {N}}_{c_{0}}(A_{l})}Cov(\overline{\mathcal {N}}_{c_{0}}(A_{l}),\overline{\mathcal {N}}_{c_{0}}(A_{j}))\nonumber \\ \end{aligned}$$

(C.20)

Also,

$$\begin{aligned} \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})}= & {} \left[ Cov^{-1}(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}))\right. \nonumber \\&+ 2\times Cov^{-1}(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{i})) \nonumber \\&+ \left. Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{i}))\right] ^{-1} \nonumber \\&\times \left[ Cov^{-1}(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}))\right. \nonumber \\&+ \left. Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}))\right] \end{aligned}$$

(C.21)

and,

$$\begin{aligned} \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}}_{c_{0}}(A_{i})}= & {} \left[ Cov^{-1}(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}))\right. \nonumber \\&+ 2\times Cov^{-1}(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{i}))\nonumber \\&+ \left. Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{i}))\right] ^{-1} \nonumber \\&\times \left[ Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}))\right. \nonumber \\&+ \left. Cov^{-1}(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{i}))\right] \end{aligned}$$

(C.22)

The fourth submatrix is:

$$ \begin{aligned}&\text {if} \ A_{i} \in M_{0} \cup M_{1}, \ \text {and} \ A_{j} \in M_{1} \ \& \ A_{j} \notin M_{0} \nonumber \\&\quad Cov\left( \tilde{\mathcal {N}}(A_{i}),\tilde{\mathcal {N}}(A_{j})\right) = \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})}Cov(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}^{\times }}_{1}(A_{j})) \nonumber \\&\quad + \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}}_{c_{0}}(A_{i})}\overline{\mathcal {N}}_{1}(A_{j}))\!\! \sum _{l}\frac{\partial r}{\overline{\mathcal {N}}_{c_{0}}(A_{l})}Cov(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{l}))~~ \end{aligned}$$

(C.23)

The fifth submatrix is:

$$ \begin{aligned}&\text {if} \ A_{i} \in M_{0} \cup M_{1}, \ \text {and} \ A_{j} \in M_{0} \ \& \ A_{j} \notin M_{1} \nonumber \\&\quad Cov\left( \tilde{\mathcal {N}}(A_{i}),\tilde{\mathcal {N}}(A_{j})\right) = \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})} \times \overline{\mathcal {N}}_{1}(A_{i}))\nonumber \\&\quad \times \sum _{l}\frac{\partial r}{\overline{\mathcal {N}}_{c_{0}}(A_{l})}Cov(\overline{\mathcal {N}}_{c_{0}}(A_{l}),\overline{\mathcal {N}}_{0}(A_{j})) \nonumber \\&\quad + \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}}_{c_{0}}(A_{i})}Cov(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}}_{0}(A_{j})) \end{aligned}$$

(C.24)

The last submatrix is:

$$\begin{aligned}&\text {if} \ A_{i} \in M_{0} \cup M_{1}, \ \text {and} \ A_{j} \in M_{0} \cup M_{1}, \nonumber \\&\quad \times Cov\left( \tilde{\mathcal {N}}(A_{i}),\tilde{\mathcal {N}}(A_{j})\right) = \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})}\frac{\partial \tilde{\mathcal {N}}(A_{j})}{\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{j})}\nonumber \\&\quad \times Cov(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{j}))\nonumber \\&\quad + \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i})}\frac{\partial \tilde{\mathcal {N}}(A_{j})}{\overline{\mathcal {N}}_{c_{0}}(A_{j})}Cov(\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{j}))\nonumber \\&\quad + \frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}}_{c_{0}}(A_{i})}\frac{\partial \tilde{\mathcal {N}}(A_{j})}{\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{j})}Cov(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}^{\times }}_{c_{1}}(A_{j})) \nonumber \\&\quad +\frac{\partial \tilde{\mathcal {N}}(A_{i})}{\overline{\mathcal {N}}_{c_{0}}(A_{i})}\frac{\partial \tilde{\mathcal {N}}(A_{j})}{\overline{\mathcal {N}}_{c_{0}}(A_{j})}Cov(\overline{\mathcal {N}}_{c_{0}}(A_{i}),\overline{\mathcal {N}}_{c_{0}}(A_{j})) \end{aligned}$$

(C.25)

1.2.3 Appendix C.2.3: Absolute normalization

In this section, we will detail the way to calculate the covariance matrix after an absolute normalization. By definition:

$$\begin{aligned} Y(A)=k\times \frac{\tilde{\mathcal {N}}(A)}{\sum _{A}\tilde{\mathcal {N}}(A)}=k\times \frac{\tilde{\mathcal {N}}(A)}{\alpha } \end{aligned}$$

(C.26)

In this work, $k=0.995\pm 0.005$. Indeed, the heavy peak is not completely measured. The slight bias has been estimated by comparing our data range with evaluated libraries. Therefore, the covariance matrix is written:

$$\begin{aligned} Cov(Y(A_{i}),\!Y(A_{j}))= & {} Y(A_{i})Y(A_{j})\frac{\sigma _{\alpha }^{2}}{\alpha ^{2}}\nonumber \\&+ Y(A_{i})Y(A_{j})\frac{\sigma _{k}^{2}}{k^{2}}\nonumber \\&- Y(A_{i})Y(A_{j})\frac{Cov(\alpha ,\tilde{\mathcal {N}}(A_{j}))}{\alpha \tilde{\mathcal {N}}(A_{j})} \nonumber \\&- Y(A_{i})Y(A_{j})\frac{Cov(\tilde{\mathcal {N}}(A_{i}),\alpha )}{\alpha \tilde{\mathcal {N}}(A_{i})}\nonumber \\&+ Y(A_{i})Y(A_{j})\frac{Cov(\tilde{\mathcal {N}}(A_{i}),\tilde{\mathcal {N}}(A_{j}))}{\tilde{\mathcal {N}}(A_{i})\tilde{\mathcal {N}}(A_{j})} \nonumber \\ \end{aligned}$$

(C.27)

with,

$$\begin{aligned} Cov(\alpha ,\tilde{\mathcal {N}}(A_{j})) = \sum _{l}Cov(\tilde{\mathcal {N}}(A_{l}),\tilde{\mathcal {N}}(A_{j})) \end{aligned}$$

(C.28)

and,

$$\begin{aligned} \sigma _{\alpha }^{2}=Cov(\alpha ,\alpha ) = \sum _{m}\sum _{l}Cov(\tilde{\mathcal {N}}(A_{m}),\tilde{\mathcal {N}}(A_{l})) \end{aligned}$$

(C.29)

Fig. 22

Comparison between common masses between Martin et al. [26] data (yellow square points), the ”Symmetry” campaign performed in September 2014 (blue diamond points), and the average between both data (red circle points) on the top left. All data are in agreement with a confidence level of 99.7$\%$. In the left bottom part, uncertainties for each data set are displayed. The new procedure allows one to reduce the uncertainty from 10 to 2$\%$. On the right, correlation matrices for each data set are displayed. All discrete masses are displayed on the matrices

1.3 Appendix C.3: Mass yield analysis: symmetry case

Similar to the regular case, the number of counts extracted from the IC are sampled using a Poisson law. We remind that for the symmetry mass region, one ionic charge distribution and one kinetic energy distribution are performed.

The first step is to extract the ionic charge probability P(q) with q the ionic charge chosen for the kinetic energy distribution. Because of the contaminants, the ionic charge distribution is chaotic as shown in Fig. 8 in the case of the mass $A=120$. Therefore, the solution is to select only the less contaminated ionic charges and perform a Gaussian fit. From the fit, we can extract the wanted P(q). This operation is performed at each sampling. Nevertheless, to be coherent with the regular mass yield analysis (where we don’t use any fit), we have tested if a Gaussian fit for the regular case is acceptable or not. In the case of masses without ns isomers [55], a Gaussian fit is acceptable only if we add a systematic uncertainty of $\varDelta _{\text {syst-q}}=7\%$ for each point. Therefore, for a symmetric mass, at each sampling, a fit is performed with this additional systematic uncertainty.

Concerning the kinetic energy distribution, two methods are developed to correct the contaminants. The extremum method is based on two free parameters $\alpha _{i}$ whereas the best estimate method is based on parameters determined through specific measurements (see Eq. (16)).

• Extremum method: at each sampling, $\alpha _{i}$ are determined by resolving the Eq. (17) and finding the higher values of $\alpha _{i}$. A Markov Chain Monte Carlo was developed to speed up the minimization. Then, the relative mass yield (at each sampling) is calculated from Eq. (18).

• Best estimate method: at each sampling, we resolve Eq. (20).

Note that we verify that the corrected kinetic distribution is always positive. If it is not the case, then a new sampling is performed. Thus, for symmetric masses, the average relative yields are calculated using Eq. (C.7). The covariance between all the masses is detailed in previously. We remind the correction from the $\left( E_{k},q\right)$ correlation can not be performed since only two distributions are measured. To take into account this deficiency, an estimator $\varDelta \mathcal {N}\left( A,q\right)$ was built and propagated on the final uncertainty. The last step is to perform the absolute normalization (for both symmetric and regular masses). This is achieved using analytical formulas (see Appendix C.2.3 for details). Finally, whereas the number of iterations is important, the CPU time is reasonable.

1.4 Appendix C.4: Complementary results

Figure 22 shows the comparison between both campaigns after the cross normalization. Both experiments are in good agreement and provide a test of the reproducibility of the LOHENGRIN spectrometer and the associated analysis. No additional uncertainty was used when averaging both experimental data. It is worth noting that the correlation of the symmetry campaign is globally positive. Indeed, such matrix indicate the weight of the systematic uncertainties over the total uncertainty. The systematic uncertainty, which correlates all the masses come from the target BU evolution. However, the mass $A=139$ has a correlation with other masses closed to 0. This is coming from the fact that for this particular mass, only one kinetic energy distribution was measured. Therefore an additional uncertainty of around 5% (see Fig. 10) was added to take into account the lack of kinetic energy distributions and cancel the correlation coming from the BU. In the case of F. Martin et al. data a more complex structure is displayed. It is coming from the fact that a self normalization procedure was already applied on this data. Finally, the merge data are highly correlated due to the cross normalization. $A=139$ is less impacted due to its larger uncertainty.

Figure 23 shows the outcome of the kinetic energy distribution corrections on the fission product mass yields. It seems that the best estimate method has a small correction on raw data. It is worth noticing that the absolute normalization is sensitive to symmetry yields. Indeed, the slight difference in yields in this region modify the sum of the measured yields. Therefore, both methods lead to a small difference in absolute yields, which are not displayed here.

Fig. 23

Comparison between the best estimate method (red circle points) and the extremum method (yellow square points) on the relative mass yields. Raw data, i.e. without any corrections (blue diamond points) are also shown

Fig. 24

Correlation matrix before (up) and after (down) normalization. These matrices reflect the analysis and the impact of the normalization

Figure 24 shows the correlation matrix before and after the normalization on the heavy peak. The correlation matrix gives information on the analysis method. It can be noticed that the positive structure is coming from the cross normalization between both Martin et al. [26] and the “Symmetry” campaign data sets. Also, correlation coming from BU is mostly washed out due to the biased estimator. Finally, the absolute normalization brings a negative correlation, as expected. The matrices displayed are the ones associated with the best estimate method analysis. This experimental matrix is different from evaluated fission yields [1, 56, 57]. For instance in the latter, paired fragments have similar correlations. This comes from the use of conservative laws in the evaluation process. In our case, we only normalize the heavy peak to 1, which explain the different behavior of paired fragments.