A new software implementation of the Oslo method with rigorous statisticaluncertainty propagation

Jørgen E. Midtbøa,1,∗, Fabio Zeisera,1,∗, Erlend Limaa,∗, Ann-Cecilie Larsena, Gry M. Tvetena,b, Magne Guttormsena,Frank Leonel Bello Garrotea, Anders Kvellestada, Therese Renstrøma

aDepartment of Physics, University of Oslo, N-0316 Oslo, NorwaybExpert Analytics AS, 0160 Oslo, Norway

Abstract

The Oslo method comprises a set of analysis techniques designed to extract nuclear level density and average γ-decaystrength function from a set of excitation-energy tagged γ-ray spectra. Here we present a new software implementationof the entire Oslo method, called OMpy. We provide a summary of the theoretical basis and derive the essential equationsused in the Oslo method. In addition to the functionality of the original analysis code, the new implementation includesnovel components such as a rigorous method to propagate uncertainties throughout all steps of the Oslo method usinga Monte Carlo approach. The resulting level density and γ-ray strength function have to be normalized to auxiliarydata. The normalization is performed simultaneously for both quantities, thus preserving all correlations. The softwareis verified by the analysis of a synthetic spectrum and compared to the results of the previous implementation, theoslo-method-software.PROGRAM SUMMARY

Program Title: OMpy [1]

Code Ocean Capsule: OMpy [2]

Licensing provisions: GPLv3

Programming language: Python, Cython

Reference of previous version: oslo-method-software

Does the new version supersede the previous version?: Yes

Reasons for the new version: Facilitate modular program flow and reproducible results in a transparent and well-documented code

base. Updated uncertainty quantification: formerly a stage-wise normalization without built-in uncertainty propagation.

Summary of revisions: Complete reimplementation; ensemble based uncertainty quantification throughout whole method; fitting

based on well-tested external libraries; corrections for the normalization procedure; auto-documentation with Sphinx

Nature of problem: Extraction of the nuclear level density and average γ-ray strength function from a set of excitation-energy

tagged γ-ray spectra including the quantification of uncertainties and correlations of the results.

Solution method: The level density and γ-ray strength function can be obtained simultaneously using a set of analysis techniques

called the Oslo method. To propagate the uncertainty from the counting statistics, we analyze an ensemble of perturbed spectra,

which are created based on the experimental input. One obtains a set of level densities and γ-ray strength functions for each

realization from a fit process. The fitting metric (χ2) is degenerate, but the degeneracy is removed by a simultaneous normalization

of the level density and γ-ray strength function to external data, such that all correlations are preserved.

Keywords: Oslo method; Nuclear level density; γ-ray strength function; Degenerate inverse problem

1. Introduction

One long-standing challenge in nuclear physics is toprecisely determine nuclear properties at excitation ener-gies above the discrete region and up to the particle thresh-old(s). This region is often referred to as the quasicon-tinuum and represents an excitation-energy range where

∗Corresponding authors: jorgenem@gmail.com (J.E. Midtbø),fabio.zeiser@fys.uio.no (F. Zeiser), erlenlim@fys.uio.no (E. Lima).

1Shared first authors.

the quantum levels are very closely spaced. That leads toa significant degree of mixing (complexity) of their wavefunctions, but they are still not fully overlapping as in thecontinuum region. For the quasicontinuum, it is fruitfulto introduce average quantities to describe the excited nu-cleus: instead of specific levels, the level density ρ(Ex) asa function of excitation energy Ex is used, and insteadof specific reduced transitions strengths B(XL) betweena given initial and final state, the average decay strengthrepresented by the γ-ray strength function (γSF) is ap-plied.

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In addition to their key role in describing fundamen-tal nuclear properties, both the level density and the γ-ray strength function are vital components for calculatingcross sections and reaction rates for explaining the nucle-osynthesis of heavy elements in astrophysics [3, 4]. Theability to calculate cross sections may also help in the de-sign of next generation nuclear reactors, where direct crosssection measurements are missing or have too high uncer-tainties for the given application [5–8].

The Oslo method [9–12] allows for extracting the leveldensity ρ and the γSF simultaneously from a data setof particle γ-ray coincidences. It has been implementedin a collection of programs known as the oslo-method-software [13], and the analysis has been successfully ap-plied to a range of nuclei in widely different mass regions[14–17]. However, the Oslo method consists of severalnon-linear steps. This makes an analytical propagationof statistical and systematic uncertainties very difficult,and thus hampers a reliable uncertainty quantification forthe final results. The statistical uncertainty propagationfrom unfolding the γ spectra and the determination of theprimary γ-ray distribution has so far not been addressedin a fully rigorous way. In lieu of this, an approximate un-certainty estimation has been used, which is described inRef. [12]. Moreover, uncertainties related to the absolutenormalization of the level density ρ and γSF have beendiscussed in Ref. [18], but there was no automatized wayto include this in the final results. Approximate ways toinclude normalization uncertainties have been suggestedand used, see e.g. Ref. [19], but they do not account forcorrelations between parameters as they were not availablewithin the oslo-method-software.

In this work, we approach the problem of uncertaintypropagation using a Monte Carlo technique. By generat-ing an ensemble of perturbed input spectra, distributedaccording to the experimental uncertainties, and propa-gating each ensemble member through the Oslo method,we can gauge the impact of the count statistics on the fi-nal results. The resulting level density and γ-ray strengthfunction have to be normalized to external data. We haveimplemented a new simultaneous normalization for bothquantities, thus preserving all correlations between them.

In the following, we present OMpy, the new implemen-tation of the Oslo method. We discuss the various stepsof the method and present our new uncertainty propaga-tion and normalization routine. The code is tested by theanalysis of a synthetic spectrum. The capability of thenew method is illustrated by applying it to experimentaldata and a comparison to the previous implementation.

2. Usage of OMpy

OMpy is designed to facilitate a more complete uncer-tainty propagation through the whole Oslo method andat the same time simplify the user interface and enhancethe reproducibility of the analysis. This section will focuson the latter two goals. OMpy, as well as all the Jupyter

notebooks and datasets used to create the figures in thisarticle, is available online [1, 2].

The documentation of the interface of OMpy is createdautomatically from its source code with the sphinx-auto-modapi package and is available from ompy.readthedocs.io.A detailed example of the usage is provided in the getting-started Jupyter notebook. Taking a step beyond justlisting the version number, we have set up a docker con-tainer to ensure also the reproducibility of the softwareenvironment in which the user runs the analysis. Thenotebooks can be run interactively online without installa-tion through Binder [20], although a limit on the availablecomputation power can lead to considerable computationtimes for cases with large ensembles. For larger calcula-tions we recommend the Code Ocean [DOI follows withpublication] capsule.

As most operations within the Oslo method will re-quire working on binned quantities like γ-ray spectra orlevel densities, or a collection of spectra into matrices, wehave created the Vector and Matrix classes. These storethe count data as one- or two-dimensional NumPy [21, 22]arrays, together with an array giving the energy calibra-tion for each axis. The classes also contain a collection ofconvenience functions e.g. for saving and loading files todisk, rebinning and plotting.

3. The Oslo method

The starting point for the Oslo method is an Ex–Eγcoincidence matrix, i.e. a set of γ-ray spectra each stem-ming from an identified initial excitation energy Ex. Astandard way to construct this input matrix is from coin-cidence measurements of γ rays and charged ejectiles fol-lowing inelastic scattering or transfer reactions with lightions. An array of γ-ray detectors measures the energy Eγof the emitted γs, while a particle telescope determines theexcitation energy Ex using the energy deposited from theoutgoing charged particles. (For a detailed description,see e.g. Ref. [18] and references therein.) Recently, othermethods have been developed that obtain the coincidencematrix from γ rays in β-decay [23] or in an inverse kine-matics setup [24]. An example of a coincidence matrix for164Dy from a standard Oslo method experiment [15, 25]at the Oslo Cyclotron Laboratory is shown in panel (a) ofFig. 1.

The first step of the Oslo method is to unfold, i.e., de-convolute the γ-ray spectra for each excitation energy tocompensate for the detector response (Compton scatter-ing, pair production, etc.). This is implemented in theUnfolder class using an iterative unfolding method de-scribed in Ref. [26]. We reiterate the main points of theprocedure in Appendix A. There we also describe a routineto select the best iteration, which has already been imple-mented in the oslo-method-software but not yet pub-lished elsewhere. The unfolded 164Dy spectrum is shownin Fig. 1 (b).

2

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Figure 1: Raw (a), unfolded (b) and first-generation (c) matrices for the 164Dy dataset [15, 25], as well as the respective relative standarddeviation matrices (d–f) obtained with the ensemble propagation technique for Nens = 50 realizations. The dashed lines in panel (c) highlightthe fit limits for ρ and f .

The second step is the determination of the first-gen-eration, or primary, γ-ray spectrum for each excitationenergy. In the FirstGeneration class an iterative proce-dure is applied as described in Ref. [27]. We recapitulatethe procedure in Appendix B, including a small additionto minimize fluctuations in higher order iterations. Theresulting first-generation γ-ray matrix is shown in panel(c) of Fig. 1. The main assumption of the first-generationmethod is that the γ-ray spectra following the populationsof levels within the excitation energy bin Ex by the inelas-tic or transfer reaction are the same as when this excita-tion energy bin is populated via γ-ray decay from higherlying levels [18]. Although this is plausible at first sight,it requires that the spin-parity distribution of the popu-lated levels is approximately independent of the excitationenergy [18, 28].

The next step of the Oslo method consists of fitting thefirst-generation spectra by a product of two one-dimen-sional functions, namely the nuclear level density ρ(Ex)

and the γ-ray transmission coefficient T (Eγ). The methodrelies on the relation

P (Ex, Eγ) ∝ ρ(Ef = Ex − Eγ)T (Eγ), (1)

where we look at deexcitations from an initial excitationenergy bin Ex to the final bin Ef , such that the γ-rayenergy is Eγ = Ex − Ef . Here, P (Ex, Eγ) is a matrix offirst-generation spectra FG(Eγ)Ex normalized to unity foreach Ex bin.

2 Furthermore, if we assume that the γ decayat high Ex is dominated by dipole radiation, the trans-mission coefficient is related to the dipole γ-ray strengthfunction f(Eγ) (or γSF) by the relation

T (Eγ) = 2πE3γf(Eγ). (2)

2Note that we follow the standard notation for the Oslo method,where P (Ex, Eγ) is the conditional probability p(Eγ |Ex) for the firstγ-ray transition with energy Eγ to come from an initial excitationenergy Ex.

3

A derivation of Eq. (1) is shown in Appendix C, where themain assumptions underpinning this decomposition are:

• The compound nucleus picture: We assume that theγ decay from an excited nuclear state is indepen-dent of how the excited state was formed. This goesback to Bohr’s theory for compound nuclei [29] andis supported by many experiments [15, 30–37].

• Dominance of dipole radiation: It is assumed thatthe decay is dominated by dipole radiation. This isstrongly supported by data and theory, see e.g. Refs.[37–48].

• The generalized Brink-Axel hypothesis: The γ-raystrength function f(Eγ) is independent of the initialand final states, i.e., it is the same for excitationsand decays between any initial and final states thatare separated by the energy Eγ [41, 49–55].

• The population cross-section: To obtain the “summed”level density ρ(Ef ) =

∑J ρ(Ef , J) in Eq. (1) the

population cross-section has to be approximately pro-portional to the intrinsic spin distribution g(Ex, J)(and we assume parity equilibration, i.e. ρ(Ex, J, π =+) ≈ ρ(Ex, J, π = −)). When the employed reac-tion is very spin selective, like in β-decay, a weightedsum of the level densities is obtained instead, see Eq.(C.14).

Finally, we have to normalize the nuclear level densityρ(Ex) and γ-ray strength function f(Eγ) (or equivalently,the γ-ray transmission coefficient T (Eγ)). It was shownby Schiller et al. [12] that the solution to Eq. (1) is in-variant under a Lie group G of transformations by threecontinuous real valued parameters A,B and α:

ρ(Ef = Ex − Eγ), f(Eγ) G→ AeαEf ρ(Ef ), BeαEγf(Eγ).(3)

However, we stress that the degrees of freedom are limitedto those given by G — i.e., all shape features of ρ and fbeyond the exponential prefactor given by G are uniquelydetermined by the fit. It is important to note that the αparameter, which influences the slope (in a log plot) of thefunctions, is common to ρ and f . Hence, their normaliza-tions are coupled together in the Oslo method. To obtainthe physical solution, the level density and γSF need to benormalized to auxiliary data. Typically, one uses s-waveresonance spacings, D0, from neutron capture experimentsas well as discrete levels to fix the level density normaliza-tion, and augment this by average total radiative widthdata, 〈Γγ〉, to normalize the γSF. This will be discussedin more detail in the next sections.

4. Uncertainty propagation by ensemble

We use an approach based on the Monte Carlo (MC)technique to estimate the statistical uncertainties in the

Oslo method by creating an ensemble of randomly per-turbed copies of the data set under study. To illustratethe method, we have chosen an experimental data set for164Dy. The data was obtained in Ref. [25] and we will com-pare our results to the reanalysis published by Renstrømet al. [15].

The random variables are the experimental number ofcounts in each bin i in the raw Ex–Eγ coincidence ma-trix R. We assume that they are independent and followa Poisson distribution with parameter λi. The Poissondistribution Pλ is given as

Pλ = p(k|λ) =λke−λ

k!(4)

We take the number of counts ki in bin i of R as an esti-mate for the Poisson parameter λi. Note that it is an unbi-ased estimator for λi, since the expectation value 〈k〉 = λ.To generate a member matrix Rl of the MC ensemble, thecounts in each bin i are replaced by a random draw fromthe distribution Pki . By this procedure, we obtain Nensmatrices representing different realizations of the experi-

ment. Defining ~ri = (r(1)i , r

(2)i , . . . , r

(Nens)i )

T as the vectorof all Nens realizations m of bin i we can calculate thesample mean 〈~ri〉,

〈~ri〉 =1

Nens

Nens∑l=1

r(m)i , (5)

and standard deviation σi,

σi =

√√√√ 1Nens

Nens∑l=1

(r

(m)i − 〈~ri〉

)2. (6)

Of course, in the case of the raw matrix R, the standarddeviation is trivial because it is given by the Poisson dis-tribution (σ =

√λ). But the technique also allows us

to estimate the standard deviation at later stages in theOslo method — after unfolding, after the first-generationmethod and even after fitting the level density and γ-raytransmission coefficient. In Fig. 1 we show the relativestandard deviations σrel,i = σi/ri in the raw (d), unfolded(e) and first-generation (f) matrices of the 164Dy datasetbased on Nens = 50 ensemble members.

It should be noted that we usually analyze spectra witha (possibly time-dependent) background. In this case wemeasure two raw spectra, the total and the backgroundspectra, which independently follow a Poisson distribution.In OMpy, both spectra can be read in and perturbed inde-pendently. The background subtracted spectra R′ are thengenerated for each realization. When the number of back-ground counts are large enough, this may lead to bins in R′

with negative number of counts. With the current default,these are removed before further processing at the cost of apotential bias towards higher level densities ρ and strengthfunctions f (see the discussion in Sec. 7). The generationof the all ensembles matrices (including application of theunfolding and first-generation method) is handled by theEnsemble class.

4

5. Decomposition into level density and γ-raystrength function

With the first-generation matrices and their standarddeviation at hand, we may proceed with the fitting of ρand T for each ensemble member. First, we select a suit-able bin size ∆E, typically 100–300 keV depending on thedetector resolution, and rebin the first-generation matrixalong both the Ex and Eγ axes to this bin size. Thematrix of experimental decay probabilities Pexp(Ex, Eγ)is obtained by normalizing the spectrum in each Ex bin tounity. For the fit of ρ and T , we take the function valuein each bin as a free parameter. For a given pair of trialfunctions (ρ, T ), the corresponding matrix Pfit(Ex, Eγ) isconstructed by

Pfit(Ex, Eγ) = CExρ(Ex − Eγ)T (Eγ), (7)

where CEx is a normalization coefficient so that∑EγPfit(Ex, Eγ) = 1 ∀Ex. We fit Pfit by a χ2 minimiza-

tion approach, minimizing the weighted sum-of-squared er-rors

χ2 =∑Ex,Eγ

(Pexp(Ex, Eγ)− Pfit(Ex, Eγ)

σPexp(Ex, Eγ)

)2. (8)

It is important to use a weighted sum rather than simplya sum of the residuals, to suppress the influence of binswith large uncertainties. This in turn makes uncertaintyestimation important. As already mentioned, a shortcom-ing of the original Oslo method implementation [12] inthe oslo-method-software has been the estimation ofthe uncertainty σPexp(Ex, Eγ) in the denominator of theχ2 fit. Due to the lack of a complete statistical uncer-tainty propagation, one has had to resort to an approx-imate uncertainty estimation. It was based on a MonteCarlo scheme similar to the present work, but where onlythe first-generation spectrum is perturbed as if each en-try was direct count data. This is discussed in detail inRef. [12]. In OMpy, we have access to a proper uncertaintymatrix σPexp . We checked that most bins of the first-generation matrices approximately follow a normal distri-bution. However, they are distributed with a larger andvarying standard deviation as compared to what one wouldhave received if the first generation entries were count datadirectly and followed the expectation value 〈k〉 = λ. Theapproximate adherence to the normal distribution justifiesthe use of a χ2 minimization as a likelihood maximization.

The χ2 minimization is carried out by numerical mini-mization in the Extractor class. This is a different imple-mentation than in the oslo-method-software, where theminimum is found by iteratively solving a set of equationsto obtain a solution satisfying ∂χ2/∂ρ = 0, ∂χ2/∂T = 0for each bin of ρ and T [12]. After testing several off-the-shelf minimizers, we have found that the modified Powell’smethod in the SciPy package works well [56, 57].

The normalized first-generation matrix P (Ex, Eγ) iscompared to the fitted matrix in Fig. 2 for one ensemble

member of the 164Dy dataset. In Fig. 3 we show the cor-responding level density ρ and γ-ray strength function f ,where the latter is obtained from T using Eq. (2).

5.1. Testing the sensitivity on the initial values

The presently used minimization routine requires aninitial guess for the trial functions (ρ, T ). In principle, thechoice of the trial functions may have a significant effecton the results if the minimizer is prone to get caught inlocal minima. Ref. [12] proposes to set the initial ρ to aconstant, arbitrarily chosen as 1, for all excitation energiesEx, and T (Eγ) as the projection of Pexp(Ex, Eγ) on theEγ-axis. In OMpy, we have implemented several choicesto test the stability of the solutions towards other initialguesses. For the cases shown here, no significant impacton the final results was observed.

Our default choice for the initial guess on ρ has beenmotivated by a long lasting discussion on whether thelevel density ρ follows the back-shifted Fermi gas (BSFG)equation [58, 59] or a constant temperature (CT) model[59, 60] below the neutron separation energy Sn , see Ref.[61, 62] and references therein. In recent years, the resultsof the Oslo method have strongly suggested a close-to CT-behavior. As this is equivalent to the initial guess for ρin Ref. [12] (after a transformation G (see Eq. (3)), weset the default initial guess to a BSFG-like solution. Morespecifically, we draw from a uniform distribution centeredaround a BSFG-like initial guess for each ensemble mem-ber. Fig. 3 demonstrates that the resulting fit is still bestdescribed by the CT-model, but it is now very unlikelythat this is due to a failure of the minimizer. The initialguess for T is still chosen as in Ref. [12], but with the samerandomization with a uniform distribution as for the leveldensity ρ.

Besides the two alternatives named above, we havealso tested a rather exotic initial guess for ρ given bya quadratic function with a negative coefficient for thesecond-degree. This contrasts any expectation that thelevel density ρ increases as a function of the excitationenergy Ex. However, even for this choice, the solution isstable.

5.2. Uncertainty estimation

Given the degeneracy of the χ2, it is nontrivial to es-timate the uncertainty in the solutions of ρ and T . Theoslo-method-software implements the approach of Ref.[12], where the uncertainty is estimated from the standarddeviation of the solutions (ρ(m), T (m)) for each realiza-tion of the first-generation matrix P

(m)exp . This may lead

to erroneously high uncertainties, as any transformation

G gives an equally good solution to a given P(m)exp . To il-

lustrate this, we could imagine that the P(m)exp ’s only differ

by a very small noise term. Due to the noise term the so-lutions will not be identical. In addition, we recapitulate

that any given P(m)exp can be equally well fit by ρ as with

any allowed transformation G such as 10ρ or 100ρ. Instead

5

of a negligible standard deviation one receives a standarddeviation solely based on the degeneracy of the solutions.So far, we have observed that the minimization results aremore stable as one might expect for the scenario outlinedhere (provided that the initial guess is not randomized).This is probably due to the way the minimizer exploresthe parameter space. Nevertheless, for the standard usageof OMpy we provide the functionality to estimate the un-certainty of solutions ρ and T after normalization of eachensemble of (ρ(m), T (m)). This will be explained in moredetail in the next section.

5.3. Fit range

Using the Oslo method we have to restrict the fit rangefor the P (Ex, Eγ) matrix. The area below E

minx exhibits

discrete transitions, thus does not adhere to the statisti-cal nature of the γ-ray strength function f and is there-fore excluded. To remain selective on the γ-decay channel,we can only use P (Ex, Eγ) up to a maximum excitationenergy Emaxx around the neutron separation energy Sn.Finally, we also constrain the minimum γ-ray energy toEminγ , which usually attributed to a deficiency of the un-folding or first-generation method for low γ-ray energies.The limits are highlighted in Fig. 1 (c) and only this validregion of P (Ex, Eγ) is sent to the minimizer.

6. Normalization of ρ and γSF

At first glance, the results bear little resemblance to alevel density or γSF. That is because the fit has not yetbeen normalized. Thus, the solution shown in Fig. 3 is justone of an infinite set of solutions to the fit. In this sectionwe will discuss how auxiliary data can be used to find thetransformation G that gives the physical solutions.

6.1. Auxiliary experimental data

For the level density ρ, there are often two differenttypes of auxiliary datasets available. At low energies, thediscrete levels are known from spectroscopy. They can becompared to the fitted level density from the Oslo methodafter applying the same binning. One can also accountfor the detector resolution by applying e.g. a Gaussiansmoothing to the histogram. At higher excitation energies,typically ∼1-3 MeV, the spectroscopy data fails to resolveall levels [61]. The user will thus have to set a sensibleregion in the low energy regime for the comparison.

The second piece of information stems from neutronresonance experiments, e.g. the average s-wave resonancespacings D0. They provide information about the leveldensity ρ(Ex =Sn, Jt ± 1/2, πt) at the neutron separationenergy Sn, where Jt and πt are the ground-state spin andparity of the target nucleus, i.e. the A − 1 nucleus [61].With the Oslo method we obtain the total level density

2000 4000 6000 8000

γ-ray energy Eγ [keV]

4500

5000

5500

6000

6500

7000

7500

Exci

tati

onen

ergy

Ex

[keV

]

(a)

2000 4000 6000 8000

γ-ray energy Eγ [keV]

4500

5000

5500

6000

6500

7000

7500

Exci

tati

onen

ergy

Ex

[keV

]

(b)

10−3

10−2

10−1

10−3

10−2

10−1

Figure 2: One realization of the normalized first-generation matrixP (Ex, Eγ) (a) compared to its fit (b) by the product of the leveldensity ρ(Ex) and γSF f(Eγ) (see Fig. 3). The dashed line indicatesthe maximum γ-ray energy Eγ = Ex. Counts to the right of thisdiagonal are due to the detector resolution or noise only and havebeen excluded from the fit. Note that panel (a) is similar to Fig. 1c,but rebinned to 200 keV and for one realization instead of the meanof the ensemble.

ρ(Ex) =∑J,π ρ(Ex, J, π). If one knows the fraction of

Jt±1/2, πt levels, one can estimate ρ(Sn) by fromD0 by [12]

ρ(Sn) =1

D0

2

g(Ex, Jt + 1/2) + g(Ex, Jt − 1/2), (9)

where g(Ex, J, π) is the spin-parity distribution of the nu-cleus at Ex and the factor of 2 comes from the assumptionof equiparity, i.e. g(Ex, J, π) = g(Ex, J)/2. Note that theJt − 1/2 term vanishes for Jt = 0. For the spin-parity dis-tribution g(Ex, I) it is common to use a form proposed inRef. [63, Eq. (3.29)]; however the exact parametrization isa major source of systematic uncertainties in the normal-ization, thus several suggestions are implemented in OMpy.

The usage of ρ(Sn) for the normalization is furthercomplicated by restricted fit-regions for P (Ex, Eγ) (see

6

0 2 4 6

Excitation energy Ex [MeV]

100

101

Lev

eld

ensi

tyρ

[arb

.u

nit

s]

(a)

result before normalization

BSFG-like initial guess

CT-like initial guess

2 4 6 8

γ-ray energy Eγ [MeV]

10−13

10−12

10−11

10−10

γS

Ff

[arb

.u

nit

s]

(b)

result before normalization

initial guess

Figure 3: Level density ρ(Ex) and γSF f(Eγ) = T (Eγ)/(2πE3γ) fromthe fit in Fig. 2. No transformation has been applied to the fit. Eventhough the initial guess for ρ was chosen from a BSFG-like function,the results are better described by the CT model above ∼ 2 MeV.The initial guess is shown before the randomization with a uniformdistribution.

Sec. 5.3). These limit the extraction of the level densityρ up to Emaxx –E

minγ , which is often about 1-3 MeV below

Sn. Consequently, we cannot directly normalize the fit-ted level density ρ to ρ(Sn) obtained from D0. To utilizethis information, we have to extrapolate ρ and comparethe extrapolation at Sn. The exact form of the extrap-olation is another systematic uncertainty. Generally, theconstant temperature (CT) model [63] fits well with thelevel density data obtained from the Oslo method [64],

ρCT(Ex) =1

Texp

Ex − E0T

, (10)

where the temperature T and the energy shift E0 are freeparameters of the model. We obtain them by a fit to ρ ina suitable energy range.

The convolution of the level density ρ and γ-ray strengthfunction f (or equivalently T ) can be further constrainedby the average total radiative width 〈Γγ(Sn)〉 from neutron-capture experiments (see e.g. Ref. [61]) on a target nucleus

with ground-state spin Jt using the following equation:

〈Γγ`(Sn)〉 =Dl2

∫ Sn0

dEγ

[f(Eγ)E

3γρ(Sn − Eγ)

×∑Ji

Ji+1∑Jf=|Ji−1|

g(Sn − Eγ , Jf )], (11)

where the first sum runs over all possible residual nucleusspins Ji, i.e. from min |Jt ± 1/2 ± `| to Jt + 1/2 + ` andthe second sum runs over all final spins Jf accessible withdipole radiation starting from a given Ji. Often only s-wave information is available, corresponding to ` = 0 in thenotation above, and the measurements are performed withlow energy neutrons, such that Ex ≈ Sn; for brevity it isthen common to write 〈Γγ〉 for 〈Γγ`(Sn)〉. A derivation ofEq. 11 is given in Appendix D. Currently, we approximate〈Γγ〉 using the integral involving the level density ρ forall excitation energies Ex, although the integral at lowenergies can be replaced by a sum over decays to the knowndiscrete levels for more precise calculations. Note that wealso have to extrapolate the γ-ray strength function to beable to use this equation. Due to its shape, a log-linearfunction is often fitted to the results.

6.2. Likelihood

In the following, we will define the likelihood L(θ) thatis used to find the proper normalizations. Let Li(θ) de-note the likelihood for a given solution (ρ, T ) of Eq. (7) tomatch the normalization information i (i.e. discrete levels,D0, 〈Γγ〉) after the transformation G with the parame-ters θ = (A,B, α). Due to the extrapolation of the leveldensity ρ mentioned above, we have to introduce two nui-sance parameters T and E0, so we extend θ to include(A,B, α, T,E0). To reduce the computational complexity,we extrapolate the γ-ray strength function by the best-fitvalues for a given set of transformations (α,B). The totallikelihood L(θ) is then given by

L(θ) =∏i

Li(θ). (12)

We assume that the experimental D0 and 〈Γγ〉 data arenormal distributed, thus maximizing the log-likelihood isequivalent to minimizing a sum of χ2i ’s. Note that the mea-surement of the discrete levels is of course not stochastic;however, the count data we use to determine ρ for the

7

comparison is. More explicitly, we have

lnLi(θ) = Ki −1

2

∑i

χ2i , (13)

χ2discrete =∑j

(ρj,discrete − ρj,Oslo(θ)

σj

)2, (14)

χ2D0 =

(D0,exp −D0,CT(θ)

σD0,exp

)2+∑j

(ρj,CT − ρj,Oslo(θ)

σj

)2, (15)

χ2〈Γγ〉 =

( 〈Γγ〉exp − 〈Γγ〉Oslo(θ)σ〈Γγ〉exp

)2, (16)

where Ki = ln(1/(2πσi)) is a constant as long as the stan-dard deviation(s) σi does not depend on θ. The subscriptexp denotes the experimental data. The sums in Eq. (14)and (15) run over all data points used in the evaluation,and for χ2D0 we invert Eq. (9) to obtain D0 from the leveldensity (extrapolated with the CT model). The secondterm of Eq. (15) arises due to the fit of the nuisance pa-rameters of the CT model (T , E0) to the Oslo methoddata, here labeled ρOslo.

As discussed in Sec. 5.2, the degeneracy of the solutionsρ and f prevents us from directly inferring their param-eter uncertainties in the fit of the first generation matrixPexp. However, clearly the data points of ρ and T havea statistical uncertainty that should propagate to informthe posterior distribution of θ. We choose to model this bysetting σj somewhat arbitrarily to a relative uncertaintyof 30%. We propose to test the implications in a futurework e.g. by comparing inferred D0’s for datasets whereD0 is known but on purpose not included in the χ

2 fit ofEq. (15). Note that with this specification of the standarddeviations σj , the Ki become θ-dependent:

Ki =∑j

ln1√

2πσj=∑j

ln1√

2π × 0.3ρj,Oslo(θ). (17)

The likelihood can easily be extended if other informationshall be taken into account. In several recent works on theOslo method, experimental data on 〈Γγ〉 was not available,but roughly estimated from systematics (see e.g. Ref. [19,65]). With the new possibilities of OMpy we would insteadrecommend to constrain α and B by adding a term tothe total likelihood that describes the match with othermeasured strength function data (usually above Sn).

6.3. Implementation

We sample this likelihood with the Bayesian nestedsampling algorithm MultiNest [66, 67] using the PyMulti-Nest module [68]. For a more efficient calculation, wefirst find an approximate solution (more accurately themaximum-likelihood estimator) θ̂ with the differential evo-lution minimizer of SciPy [56, 69]. This is by default used

to create weakly informative priors for A, B and α and T .For A and B we use a normal distribution truncated at 0(negative values of ρ or f are not meaningful), a defaultmean µ given by θ̂ and a broad width of 10µ. For α and thenuisance parameter T (entering through the level densityextrapolation model) we use log-uniform priors spanningone order of magnitude around θ̂. For the second nuisanceparameter E0 we choose a normal distribution with mean0 and width 5 MeV, which is truncated below (above) -5and 5 MeV, respectively. The latter choice is well justifiedregarding the range of reported values of E0 in Ref. [70].

This simultaneous normalization is implemented in theNormalizerSimultan class, which relies on composition ofthe NormalizerNLD and NormalizerGSF classes handlingthe normalization of the level density ρ and γ-ray strengthfunction f . To facilitate a comparison with the oslo-method-software calculations, one can also run a sequen-tial normalization first using Eq. (14) and (15) throughNormalizerNLD and then the resulting ρ as input to thenormalization through NormalizerGSF, see Eq. (11). Itshould be stressed though that the normalization throughthe likelihood calculations in OMpy will still differ from theapproach taken in the oslo-method-software. The latterallowed only to receive best-fit estimates of the transforma-tion parameters A, B and α. Any subsequent uncertaintyestimation due to the normalization itself was up to theusers. We also note that an advantage of the simultane-ous approach is that one obtains the correlations betweenA, B and α such that uncertainties in the normalizationof ρ directly propagate to the estimation transformationparameters for f .

From the MultiNest fit we obtain the posterior proba-bility distribution for the parameters θ, given as (equally-weighted) samples θi.

3 The normalization uncertainty forthe solution (ρ(m) , f (m)) of the realization m can then bemapped out by creating a normalized sample (ρ(m), f (m))ifor each θi using Eq. (3). By repeating this procedure forall realizations m of the ensemble, we also recover the un-certainty due to the counting statistics. This is performedin the EnsembleNormalizer class.

7. Systematic uncertainties

The previous sections described the necessary tools toevaluate the statistical uncertainties due to the countingstatistics and the normalization procedure. It is importantto keep in mind that there are also systematic uncertaintieslinked to the analysis, which are summarized below:

a) Removal of negative counts in raw matrix : When sub-tracting the background from the raw matrix, we of-ten receive a matrix with negative counts in some bins.This is clearly linked to the Poisson statistics in re-gions with a low signal to background ratio. If one

3For brevity, we drop the index m on θ, but the normalization isperformed for each realization m individually.

8

simply removes the negative counts, one potentially bi-ases the mean of the level density and γ-ray strengthpoints derived from these bins. It was previously ob-served that negative counts in the raw matrix can causetechnical challenges in the currently implemented un-folding method, with some bins obtaining extreme neg-ative values after several iterations. For the backgroundsubtraction, it might be a more reasonable fix to redis-tribute the negative counts to bins within the resolu-tion. This is implemented by Matrix.fill negativeas an alternative to the default method, Matrix.remove -negative, which removes all negative counts. For the164Dy, there is a high signal to background ratio, thusthe background subtraction does not have this problem.In cases with a worse background ratio, both methodscould be compared. If they result in significant dif-ferences, further analysis is needed to find the optimalprocedure.

b) Removal of negative counts in unfolded and first-gener-ation matrix : The unfolding and first-generation meth-ods can result in negative counts which can not belinked to the Poisson statistics any longer. It is thus notclear whether it is better to keep the negative countsor to redistribute them in the fashion described above.Again, the problem was negligible in the 164Dy dataset,but has to be treated carefully if the methods lead tomore bins with such a behavior. To retrieve the ma-trices before removal of the negative counts, the usercan simply replace or deactivate the remove negativemethods of the Unfolder and FirstGeneration classes.

c) Unfolding method : There are two main sources of sys-tematic uncertainties, the iterative unfolding methoditself, and the detector response functions. The lattercan be gauged by obtaining an ensemble of differentdetector response functions that capture the breadthof physically reasonable configurations (e.g. auxiliarysoftware such as GEANT4 [71]). Each member of theraw matrices R(m) is then unfolded with one (or each)of the different detector response functions. The formeruncertainty is more difficult to quantify. In this spe-cial case, alternative methods exist already and one ap-proach could be to implement an alternative unfoldingalgorithm (e.g. Ref. [72]). The oslo-method-softwareattempts to quantify the systematic uncertainty fromunfolding and the first-generation matrix following anad hoc numerical procedure described in Ref. [12]. Aproper treatment that pays tribute to the full complex-ity of the problem is outside the scope of the presentwork.

d) Population cross-section: As mentioned in Sec. 3, thefirst-generation method assumes that the spin-paritydistribution of the populated levels gpop is similar forthe whole excitation energy range studied. At best,gpop approximates the spin-parity distribution g of thelevels in the nucleus itself. This is often believed to be

the case for low and mid-mass nuclei, where the beamenergy is chosen such that the compound cross-sectiondominates over the direct cross-section. More detailscan be found in Refs. [18, 73] for challenges if the in-trinsic spin-parity distribution g(Ex, J, π) is very dif-ferent from the (normalized) population cross-sectiongpop(Ex, J, π).

e) Decomposition and the Brink-Axel hypothesis: The de-composition of the first-generation matrix into the leveldensity ρ(Ex) and γ-ray strength function f(Eγ) re-lies on a generalization of the Brink-Axel hypothesis[49, 50], where the strength function is assumed to beapproximately independent of Ex, J and π (see Eq.(C.5)). The validity of the assumption has been testedwithin the Oslo method by comparison of the γ-raystrength function f extracted from different initial andfinal excitation energy bins, see Refs. [51, 52] and ref-erences therein for further works.

f) Intrinsic spin-parity distribution: Both the normaliza-tion of the level density ρ at Sn via D0 and the absolutescaling of the γ-ray strength function via 〈Γγ〉 rely onthe knowledge of the intrinsic spin-parity distributiong(Ex, J, π) of the nucleus, see Eq. (9) and Eq. (11).It is difficult to measure the spin-parity distributiong(Ex, J, π) in the (quasi)continuum and there are vari-ous different empirical parametrization and theoreticalcalculations, see e.g. Refs. [59, 70, 74, 75] and Refs.[76–79] respectively.

g) Lack of D0 and 〈Γγ〉: In several recent cases wherethe Oslo method has been applied, experimental val-ues of D0 and/or 〈Γγ〉 were not available, see e.g. Refs.[15, 19, 37, 65]. In these works, D0 and/or 〈Γγ〉 wereestimated from the values of nearly nuclei. With OMpyit is now easier to adopt the normalization procedurevia the likelihood L(θ) in Eq. (12) instead. One can ei-ther remove terms where the experimental informationis missing, allowing for a larger degeneracy of the solu-tions, or add different constrains, like a measure of howwell the γ-ray strength function f(Eγ) matches othermeasured γ-ray strength function data, which often ex-ists above Sn.

h) Bin sizes and fit ranges: Several decisions have to bemade on the bin sizes and fit ranges, for example inthe normalization of the level density ρ to the knownlevel scheme, see Sec. 6.1. The user can test differentsensible regions for this comparison.

In this work, only the impact of the negative counts,items a and b have been studied. In many cases it ischallenging to evaluate the impact of the assumptions onthe data. Whenever the uncertainty is linked to models,algorithms, parameter sets or alike, and alternative choicesexists (items c, d, f, h), it is possible to utilize OMpy’serror propagation functionality. This was illustrated for

9

the unfolding of the data in item c. More information onitems e and f can be found in Ref. [18].

8. Discussion and comparison

In Fig. 4 we show the level density ρ and γ-ray strengthfunction f of 164Dy resulting from the simultaneous nor-malization of Nens = 50 realizations. Each realization m istransformed with Nsamp = 100 samples from the normal-ization parameters θi. The combined uncertainty of thenormalization and counting statistics is visualized throughthe 16th, 50th, and 84th percentiles which together formthe median and a 68% credible interval. Additionally, weshow one randomly selected sample (ρ(m), f (m))i includingits extrapolation. The results are compared to the analy-sis of Renstrøm et al. [15] which used the oslo-method-software, and we display both the uncertainty that isquoted due to the counting statistics and the total un-certainty, that includes the normalization. Note that wediscussed in Sec. 5.2 that this split into the uncertainty dueto counting statistics and due to normalization is, strictlyspeaking, not possible – hence, the quoted decompositionis an approximation.

It is gratifying to see that overall, both analyses providesimilar results. In Fig. 4a, the level density below ∼ 2 MeVexhibits the same structure of bumps attributable to thediscrete level structure, and at higher Ex the curves arepractically identical. Similarly, for the γSF in Fig. 4b, thedata is mostly compatible within the error-bars. The me-dian of the results from OMpy has slightly steeper slope andthe uncertainties are somewhat more evenly distributedacross the whole energy range.

Some deviation between the results is expected due tothe different fitting method and software implementations.Moreover, different fit-regions may lead to different estima-tions of the normalization parameters θ, which in turn givedifferent slopes (and absolute values) for ρ and the γSF.Usually, there will be several sensible fit-regions. WithOMpy, the user could create a wrapper that loops throughthe different fit-regions. The mean and spread of the re-sults can be analyzed using the same ensemble based ap-proach as above.

Another way to verify the results of OMpy is to simulatedecay data for a given ρ and the γSF and compare theseto the analysis with OMpy from the simulated data. Wehave used the Monte-Carlo nuclear decay code RAINIERv1.5 [81, 82] to create decay data from a 164Dy-like nu-cleus. To create an artificial level scheme, we used the first20 known discrete levels [61] and for higher energies cre-ated levels following the CT model and spin-distributiondescribed in Renstrøm et al. [15]. The γSF has been mod-eled with the parameters from the same publication. Then,we simulate the experiment by populating 2 × 106 levelsbelow Sn and recording the decay γs for each event. Forsimplicity, we have assumed here that we populate thelevels proportionally to the intrinsic spin distribution ofthe nucleus. Finally, we use the response functions of the

164Dy experiment to convert the recorded γ rays to a ma-trix of synthetically generated events that substitute theexperimentally determined raw matrix in the further anal-ysis with OMpy. The full setting file can be found with thesupplementary material online.

In Figure 5 we compare the results of OMpy to the inputlevel density ρ and γSF from RAINIER. The fitted leveldensity ρ slightly over-pronounces the structures of thediscrete levels at low excitation energies. From a practicalpoint of view this is not a significant problem, as the Oslomethod is used for an analysis of the quasicontinuum re-gion, so at energies & 2 MeV. There, we observe a perfectmatch between the input and the fitted level density. Theresulting γSF is in very good agreement with the inputγSF. There are small deviations at the lowest and high-est energies that can easily be understood. The apparentdiscrepancy of the input γSF below ∼ 2 MeV can be ex-plained by a failure of the first-generation method due tostrong populations of discrete states and is directly visi-ble in the comparison of the unfolded to first-generationmatrix (see supplementary material online). The regioncould have been excluded from the comparison, but weaimed for the same extraction region as for the experi-mental 164Dy data set. The mismatch propagates to awrong extrapolation of the γSF towards lower energies.The two resonance-like structures at ∼6 and 7 MeV over-compensate for the mismatch of the level density at thelowest excitation energies.

One of the main motivations for OMpy was to improvethe uncertainty analysis in the Oslo method. Throughoutthe article we have highlighted several improvements tothe theoretical framework for propagating the uncertain-ties compared to the oslo-method-software. The mostfundamental source of uncertainties in the analysis is thecount statistics. For each experiment a balance betweenthe run-time costs and the possibility to gather more datahas to be found. Therefore, we find it instructive to studythe impact of a reduced number of counts for the 164Dyexperiment. We create a new raw matrix, which we drawfrom a Poisson distribution with a mean of 1/10th of theoriginal data (an arbitrary choice corresponding to 1/10thof the run actual time) and otherwise perform the sameanalysis as above. Both analyses are compared in Fig. 6,and overall we find a good agreement of the median values.

There is one major difference in the analysis that leadsto different uncertainty estimates. In the case with the re-duced number of counts we have to use a lower maximumexcitation energy Emaxx in the fit to P (Ex, Eγ). As a conse-quence, the extracted level density ranges only up to about4.5 MeV instead of 6.5 MeV, increasing the uncertainty inthe determination of the normalization parameters. Theincreased uncertainty of the slope parameter α is directlyvisible for the level density ρ. For the γSF we addition-ally require a suitable fit to the experimental 〈Γγ〉, so adifferent slope α will also affect the absolute scaling B.This results in the normalized γSFs tilting around a γ-rayenergy of about 5 MeV, where the exact location of the

10

https://doi.org/10.24433/CO.6094094.v1https://doi.org/10.24433/CO.6094094.v1

0 2 4 6 8

Excitation energy Ex [MeV]

101

102

103

104

105

106

Lev

eld

ensi

tyρ(E

x)

[MeV−

1]

(a)

results

random sample

model

fit limits

known levels

ρ(Sn)

Renstrøm et al.

0 2 4 6 8

γ-ray energy Eγ [MeV]

10−8

10−7

γ-S

Ff(Eγ)

[MeV−

3]

(b)

Figure 4: Extracted level density ρ (a) and γSF f (b) for 164Dy. The fit is similar to that shown in Fig. 3, but the normalization accordingto Eq. (3) has been applied. We display the median and 68% credible interval obtained from the counting and normalization uncertainties(orange line and blue shaded band, respectively) together with one randomly selected sample of ρ and f (blue dots). The median and 68%credible interval for the extrapolation is given by the green line and shaded band. In addition, the extrapolation used together with therandom sample is indicated by the dashed line. The data points within the gray area denoted fit limits are used for the normalization andextrapolations, such that we match the binned known levels (black line) [80], ρ(Sn) calculated from D0 [61] (and 〈Γγ〉 [61]). The results arecompared to the analysis of Renstrøm et al. [15] which used the oslo-method-software, displaying both the uncertainty that is quoted dueto the count statistics (inner error bar) and the total uncertainty, including the normalization (outer error bar) (see text for more details).

tilting point depends on the nucleus. The normalizationuncertainty thus leads to increasing relative uncertaintiesof the γSF for lower γ-ray energies Eγ . This contrasts tothe naive expectation that the relative uncertainties shouldincrease with Eγ , as the number of counts that determineeach γSF bin decrease. Finally, we foresee that the up-dated framework for the uncertainty analysis may have asignificant impact when further processing of the resultsfrom the Oslo method. An example of this is given inRenstrøm et al. [15], where the authors fit of the strengthof the peak at ∼ 3 MeV in the γSF which assumed to bea M1 scissors mode.

9. Extension of OMpy

OMpy is written with modularity in mind. We want it tobe as easy as possible for the user to add custom function-ality and interface OMpy with other libraries. For example,in Sec. 7 we discussed that it may be of interest to tryother unfolding algorithms than the one presently imple-mented. To achieve this, one just has to write a wrapperfunction with the same return structure as the callableUnfolder class. Then one provides the new wrapper toEnsemble instead of the Unfolder class and all matriceswill be unfolded with the new algorithm.

It is our hope and goal that OMpy will be used as muchas possible. Feedback and suggestions are very welcome.We encourage users who implement new features to sharethem by opening a pull request in the Github repository.

10. Conclusions and outlook

We have presented OMpy, a complete reimplementationof the Oslo method in Python. The capabilities of the codehave been demonstrated by comparison with a syntheticdata set modeled with the decay code RAINIER. We havealso compared OMpy to the analysis for 164Dy with theprevious implementation of the Oslo method, the oslo-method-software and in general find a good agreement.However, we have refined the uncertainty quantification ofthe analysis using an ensemble-based approach. We arenow able to simultaneously take into account statisticaluncertainties from the counting statistics and the normal-ization to external data and preserve the full correlationbetween the resulting level density ρ and γSF.

Acknowledgements

We would like to thank V. W. Ingeberg for a beta testand his feedback on the software. This work was sup-ported by the Research Council of Norway under project

11

https://github.com/oslocyclotronlab/ompy

0 2 4 6 8

Excitation energy Ex [MeV]

101

102

103

104

105

106

Lev

eld

ensi

tyρ(E

x)

[MeV−

1]

input

results

random sample

model

fit limits

known levels

ρ(Sn)

0 2 4 6 8

γ-ray energy Eγ [MeV]

10−8

10−7

γ-S

Ff(Eγ)

[MeV−

3]

Figure 5: Extracted level density ρ (a) and γSF f (b) from synthetic data for a 164Dy-like nucleus. In the quasicontinuum region the analysiswith OMpy (labeled results) agrees well with the input level density and γ-ray strength function model.

Grants No. 263030, 262952 and 23054. A.C.L. gratefullyacknowledges funding from the European Research Coun-cil, ERC-STG-2014 Grant Agreement No. 637686.

Appendix A. Unfolding

Here we explain the unfolding technique presented inRef. [26], which is used both in the original Oslo methodimplementation in the oslo-method-software and in OMpy.Let the detector response be modeled as a conditionalprobability density function

p(Eγ |E′γ), (A.1)

encoding the probability that a γ ray with true energy E′γis detected with energy Eγ . Given a true γ-ray spectrumU(Eγ), the folded spectrum F (Eγ), i.e. the spectrum seenby the detector, is then given by

F (Eγ) =

∫p(Eγ |E′γ)U(E′γ) dE′γ . (A.2)

By discretizing into energy bins of width ∆Eγ , it becomesa matrix equation

~F = P ~U, (A.3)

where P is the response matrix of discrete probabilitiesPkl = p(Eγ,k|E′γ,l)∆Eγ . The unfolding procedure amountsto solving this equation for ~U . However, a straightfor-ward matrix inversion is ill-advised, as it will often lead tosingularities or produce large, artificial fluctuations in ~U

[83, 84]. Instead, the approach taken in the Oslo methodis to use an iterative technique that successively approx-imates ~U . Letting ~R denote the measured spectrum, thealgorithm is

1. Start with a trial function ~U0 = ~R at iteration i = 0

2. Calculate the folded spectrum ~Fi = P ~Ui

3. Update the trial function to ~Ui+1 = ~Ui + (~R− ~Fi)

4. Iterate from 2 until ~Fi ≈ ~R.

Note that the oslo-method-software uses a custom tai-lored combination of the additive updating procedure ofstep 3, and a ratio approach, ~Ui+1 = ~Ui ◦ (~R� ~Fi), wherethe Hadamard products stand for an element wise prod-uct. We obtain equally good results by adopting only theadditive updating in OMpy. The updating procedure maysometimes lead to a negative number of counts in the un-folded spectra. For negative counts close to zero, it is notclear whether these should be kept or discarded, as it isnot clear whether they originate from the statistical na-ture of the data. In some cases, one observes some binswith large, negative counts which hints at a failure of themethod. These cases should be analyzed carefully beforethe results are processed further. The current default is toremove the negative counts at the end of the unfolding.

A too large number of iterations does not improve theresults significantly, but introduces strong fluctuations inthe unfolded spectrum. After the publication of Ref. [26]

12

0 2 4 6 8

Excitation energy Ex [MeV]

101

102

103

104

105

106

Lev

eld

ensi

tyρ(E

x)

[MeV−

1]

× 1/5

(a)

results

random sample

model

fit limits

known levels

ρ(Sn)

0 2 4 6 8

γ-ray energy Eγ [MeV]

10−8

10−7

γ-S

Ff(Eγ)

[MeV−

3]

× 1/5

(b)

Figure 6: Similar to Fig. 4 we show the extracted level density ρ (a) and γSF f (b) for the 164Dy experiment, but here it is compared to acase with only 1/10th of the data. The level density ρ (a) and γSF from the case with reduced number of counts have been scaled down bya factor of 5 in the plot to facilitate the visual comparison with the original analysis. For each case, one randomly selected sample includingits extrapolation is shown in addition to the median and a 68% credible interval.

a criterion for step 4 has been added to the oslo-method-software, which is also used in OMpy. A predefined num-ber (usually around 30-200) iterations is run and the bestiteration is selected based on a weighted sum over eachvector element of the root-mean-square error of ~Fi − ~Rand the relative level of fluctuations in ~Ui compared tothe fluctuations of the raw spectrum ~R. The relative fluc-tuations are estimated as |(~Ui,l − 〈 ~Ui〉)/〈 ~Ui〉|1, where 〈 ~Ui〉is a smoothed version of the spectrum ~Ui.

In addition to this, the user can choose to use a furtherrefinement to the unfolding method known as Comptonsubtraction [26]. It is used to further control the fluctua-tions in the unfolded spectrum. The basic concept behindit is to use the previously unfolded spectrum to decom-pose ~R into parts corresponding to the full-energy, singleand double escape and annihilation peaks, and the “rest”which comes from Compton scattering and similar pro-cesses. Each of these parts, save for the full-energy peak,are then smoothed with the detector resolution before theyare subtracted from ~R. The resulting spectrum normalizedto maintain the number of counts. The idea is that thisshould give an unfolded spectrum with the same statisticalfluctuations as in the original spectrum ~R.

Appendix B. The first-generation method

In this appendix we describe the idea behind the first-generation method of Ref. [27] and its implementation inOMpy. Let FG(Eγ)Ex denote the first-generation γ-ray

spectrum, i.e., the intensity distribution of γ-ray decayfrom a given excitation energy Ex, as function of γ-rayenergy Eγ . Generally, the nucleus will decay from Exdown to the ground state by emitting a cascade of γ rays,which forms the total γ-ray spectrum. The total, or all-generations γ-ray spectrum, denoted AG(Eγ)Ex , can beviewed as a superposition of the first-generation spectrumand a weighted sum of the all-generations spectra of exci-tation energies below,

AG(Eγ)Ex = FG(Eγ)Ex (B.1)

+∑

E′x

thus the number of γ-ray cascades out of this level,

n(E′x)Ex =S(Ex)

S(E′x). (B.2)

However, usually it is determined from the total γ-rayspectrum by the relation

n(E′x)Ex =〈M(E′x)〉N(Ex)〈M(Ex)〉N(E′x)

, (B.3)

where 〈M(Ex)〉 and N(Ex) denote the average γ-ray mul-tiplicity and the total number of counts, respectively, atexcitation energy Ex. This reformulation uses the fact thatS(Ex) = N(Ex)/M(Ex). In OMpy there are two ways todetermine the average multiplicity 〈M(Ex)〉. The initialidea, the total multiplicity estimation is given in Ref. [9]as

〈M(Ex)〉 =Ex〈Eγ〉

, (B.4)

where 〈Eγ〉 is the weighted-average γ-ray energy at ex-citation energy Ex. Due to the detector threshold, weare usually not able to measure all γ rays, and this willartificially increase 〈Eγ〉. To solve this problem, a statis-tical multiplicity estimation has been added to the oslo-method-software and is adapted in OMpy. The underlyingidea is that in heavier nuclei, like in the rare-earth region,γ rays from entry states at higher excitation energy willdecay down to the yrast-line, where it enters at an energydenoted here as Eyrast. From there, the γ rays follow anon-statistical decay to the ground state. For heavy nu-clei, there are many levels at low excitation energies, suchthat it is assumed here that the yrast transitions will pro-ceed with many γ rays of so low energy, that they areusually below the detector threshold. In that case, we canreplace the excitation energy in Eq. (B.4) by the apparentexcitation energy Ẽx = Ex − Eyrast. A challenge in thismethod is to correctly estimate the entry energy Eyrast.Thus we recommend to use the total multiplicity estima-tion whenever the experimental conditions allow for it.

Weight function and iteration

The weight function w(E′x) encodes the probability forthe nucleus to decay from Ex to E

′x, and is in fact nothing

but the normalized first-generation spectrum for Ex,

w(E′x)Ex =FG(Ex − E′x)Ex∑

E′γFG(E′γ)Ex

. (B.5)

By rewriting Eq. (B.2), we obtain

FG(Eγ)Ex = AG(Eγ)Ex (B.6)

−∑

E′x

channel. In the Oslo method, we select only excitation en-ergies Ex below the particle separation threshold, so thecompound nucleus can decay by γ-rays only. The decayprobability GCN of the state i is then simply the γ-raybranching ratio to a specific final state f with energy Efand the spin-parity Jπf ,

GCN =Γγ(i→ f)

Γγ, (C.2)

where Γγ(i → f) is a partial, and Γγ =∑f Γγ(i → f) is

the total radiative width for the level i.The spectrum of the decay radiation n(i, f)dEf from

the compound nucleus level i is then given by the summa-tion (or integration) of σ(α, f) over an interval dEf (upto a constant due to the flux of a and density of A whichwould cancel out later):

n(i, f)dEf =∑

f in Ef

σ(α, f) = σCN(α)∑

f in Ef

Γγ(i→ f)Γγ

= σCN(α)∑XL

〈Γγ(i→ f)〉Γ

(L)γ

ρavail(Ef ),

= σCN(α)∑XL

fXL(Eγ)E2L+1γ

Γ(L)γ ρ(Ex, Jπi )

ρavail(Ef )

= Cα,i∑XL

fXL(Eγ)

Γ(L)γ

E2L+1γ ρavail(Ef ), (C.3)

where ρavail is density of accessible final levels at Ef , Cα→iis a constant that depends only on the entrance channel(which determines the compound nucleus state i), and wehave replaced the average partial radiative width 〈Γγ(i→f)〉 by γ-ray strength-function5 fXL.

The γ-ray strength function fXL for a given multipo-larity XL and for decays from an initial level i to finallevel f is defined as [89]

fXL(Ex, Jπi , Eγ , J

πf ) =

〈ΓγXL(Ex, J

πi , Eγ , J

πf

〉E2L+1γ

ρ(Ex, J, π)

(C.4)

where 〈· · · 〉 denotes an average over individual transitionsin the vicinity of Ex (in practice defined by the energybinning resolution). This can be simplified using the dom-inance of dipole radiation (L = 1) and a common gener-alization of the Brink-Axel hypothesis [49, 50], where thestrength function is assumed to be approximately indepen-dent of Ex, J and π,∑

XL

fXL(Ex, Jπi , Eγ , J

πf )

dipole≈ f1(Ex, Jπi , Eγ , Jπf )

Brink-Axel≈ f1(Eγ) (C.5)

5To keep standard notation we will denote both the γ-ray strengthfunction and the final level by f . It should be clear from the contextwhat we refer to.

where we define the total dipole strength function f1 asthe sum of the electric and magnetic dipole strength, fE1and fM1, respectively, f1 = fE1 + fM1.

If we again use the dominance of dipole radiation in

Eq. (C.3) (which leads to Γγ ≈ Γ(L=1)γ ), and assume par-ity equilibration of the level density, i.e. ρ(Ex, J,+) ≈ρ(Ex, J,−) we can write

n(i, f)dEf ≈ Cα,iE3γ

×

fE1(Eγ)Γγ

Jf=Ji+1∑Jf=Ji−1

ρ(Ef , Jf ,−πi)

+fM1(Eγ)

Γγ

Jf=Ji+1∑Jf=Ji−1

ρ(Ef , Jf , πi)

= C ′α,iE

3γf1(Eγ)

Jf=Ji+1∑Jf=Ji−1

ρ(Ef , Jf , eq), (C.6)

where ρ(Ex, Jf , eq) denotes the level density of one parity,the notation emphasizing the assumption of parity equili-bration6 and C ′α,i = Cα,i/Γγ . We may write the partiallevel density ρ(Ex, J, eq) as

ρ(Ex, J, eq) =1

2g(Ex, J)ρ(Ex), (C.7)

where g denotes the intrinsic spin distribution of the nu-cleus and ρ(Ex) =

∑Jπ ρ(Ex, J, π) is the “summed” (or

“total”) nuclear level density. With this, we can furthersimplify the sum over the final levels in Eq. (C.6):

Jf=Ji+1∑Jf=Ji−1

ρ(Ef , Jf , eq) (C.8)

=ρ(Ef )

2

Jf=Ji+1∑Jf=Ji−1

g(Ef , Jf ) ≈3ρ(Ef )

2g(Ef , Ji),

which is a good approximation except for the case of Ji = 0(and Ji = 1/2), where the selection rules allow only transi-tions to Jf = 1 (Ji = {1/2, 3/2}) states.

Next, we write n(i, f)dEf more explicitly as I(Ei, Jπi , Eγ)

and exploit probability conservation,

PJπi (Ei, Eγ) =I(Ei, J

πi , Eγ)∑

EγI(Ei, Jπi , Eγ)

(C.9)

= CEi,Jπi E3γf1(Eγ)ρ(Ei − Eγ)g(Ei − Eγ , Ji)

where PJi,πi(Ei, Eγ) is the probability to decay from aninitial excitation energy bin Ei with a γ ray of energy Eγ ,

6The selection rules dictate that dipole radiation changes the an-gular momentum J by at most one unit. For M1, the parity isunchanged, while for E1 it flips. This determines the density ofavailable final levels for the decay. In the case of Ji = 1/2 the sumruns over Jf = {1/2, 3/2}, and in the case of Ji = 0, the sum onlyruns over Jf = 1, since J = 0→ J = 0 transitions are forbidden.

15

the subscripts Ji and πi limit the initial levels to of a givenspin and parity, respectively, and CEi,Jπi is a normalizationconstant. Note that the normalization constant C ′α,i can-cels out (which includes compound nucleus cross-sectionσCN, the total radiative width Γγ , and the density of in-trinsic levels ρ(Ei, J

πi )).

The final step is to generalize this equation for the casewhere levels of different spins and parities are populated.Naively, we may just sum over the decays PJπi from allinitial levels Jπi∑

Jπi

PJπi ≈ CEiE3γf1(Eγ)ρ(Ei − Eγ)

∑Jπi

g(Ei − Eγ , Ji)

= CEiE3γf1(Eγ)ρ(Ei − Eγ), (C.10)

from which we would already recover the standard Oslomethod equation for a suitable normalization constant CEi .However, we have to note that the normalization constantsCEi,Jπi in Eq. (C.9) depend on the spin and parity, andcannot be factored out. As we will see, this can be solvedunder the assumption that the cross-section σCN to createthe compound nucleus at different spin-parities Jπi is pro-portional to number of levels in the nucleus (i.e. it is notspin-selective, but proportional to intrinsic spin distribu-tion),

σCN(α→ Ei, Jπi ) ≈ σCN(Ei)ρ(Ei, Ji, πi)= σCN(Ei)ρ(Ei)g(Ei, Ji, πi). (C.11)

Using Eq. (C.6) to (C.8), the (cross-section weighted) sumover the decay spectra from all populated levels is

I(Ex, Eγ) =∑Jπi

I(Ex, Eγ , Jπi )

=∑Jπi

σCN(Ei)ρ(Ei, Ji, πi)

Γγρ(Ei, Ji, πi)f1(Eγ)E

3γρavail(Ef )

≈ σCN(Ei)Γγ

f1(Eγ)E3γ

3ρ(Ef )

2

∑Jπi

g(Ex, Ji)

= 3σCN(Ei)

2Γγf1(Eγ)E

3γρ(Ex − Eγ). (C.12)

In principle, we also have to average over the excitationenergy bin Ex. However, as the level density ρ and thetotal average radiative width Γγ are assumed to vary onlyslowly with energy, this will not lead to any changes in theequation above. The normalized spectrum is given by

P (Ex, Eγ) =I(Ex, Eγ)∑EγI(Ex, Eγ)

= CExf1(Eγ)E3γρ(Ef = Ex − Eγ), (C.13)

for a normalization constant CEx that depends only on theexcitation energy.

In the case of a more spin-selective population of thecompound nucleus, like in β-decay one receives a weighted

sum of the level densities. If we denote the normalizedpopulation of the levels per for each excitation energyby gpop(Ex, J, π) = σCN(Ex, J, π)/

∑Jπ σCN(Ex, J, π), Eq.

(C.12) can be generalized as

P (Ex, Eγ) (C.14)

= CExf1(Eγ)E3γρ(Ef = Ex − Eγ)

∑Jπ

gpop(Ex, J, π)

g(Ex, J).

In summary, we have shown that the Oslo methodequation arises naturally from the Bohr’s independence hy-pothesis for the compound nucleus under the assumptionof i) the dominance of dipole radiation, ii) parity equili-bration of the level density and iii) a compound nucleuscross-section σCN that is proportional to the spin distribu-tion of intrinsic levels.

Appendix D. Calculation of 〈Γγ〉

In this appendix we derive Eq. (11) used to calculatethe average total radiative width 〈Γγ〉 from the level den-sity ρ and γ-ray strength function f . The derivation isbased on Eqs. (7.19-7.23) in Ref. [87] and p. 106 in Ref.[50]. It bases on the same arguments as Appendix C, butfor ease of readability, we reiterate the main points.

The radiative width Γγ denotes the probability for astate to decay by γ-ray emission. In the multipolar expan-sion it is written as

Γγ =∑L

(ΓEL + ΓML), (D.1)

where EL and ML denote the electric and magnetic com-ponents of the radiation of order L, respectively. To sim-plify the derivation, we will now assume the dominance ofdipole radiation, such that

Γγ = ΓE1 + ΓM1. (D.2)

We will continue with the derivations for the E1 radia-tion, but similar equations hold for M1. Analogously toEq. (C.3), we can express ΓE1 as the sum of the partialdecay widths ΓE1,i→f from the initial level i to allowedfinal levels f ,

ΓE1 = Γγ,E1(Ex, Ji, πi) =∑f

ΓE1,i→f , (D.3)

where the initial level is at the excitation energy Ex andhas the spin Ji and parity πi. The angular momentum andparity of the final states are given by the usual selectionrules.

We now rewrite the partial widths ΓE1,i→f in termsof the strength function f , using the definition of f , Eq.(C.4), and the generalized Brink-Axel hypothesis, Eq. (C.5),

ΓE1,i→f → 〈ΓE1(Eγ , Ex)〉 =fE1(Eγ)E

3γ

ρ(Ex, Ji, πi)(D.4)

16

where ρ(Ex, Ji, πi) is the density of the initial levels, andthe energy difference to the final state(s) is given by Eγ =Ex − Ef .

Next, we replace the the sum in Eq. (D.3) by an in-tegral, where we note that the number of partial widths〈ΓE1(Eγ , Ex)〉 is proportional to the level density at thefinal states ∑

Jf

∑πf

ρ(Ex − Eγ , Jf , πf ), (D.5)

such that the average total radiative width 〈Γγ,E1(Ex, Ji, πi)〉is given by

〈Γγ,E1(Ex, Ji, πi)〉 (D.6)

=

∫ Ex0

dEγ 〈ΓE1(Eγ , Ex)〉∑Jf

∑πf

ρ(Ex − Eγ , Jf , πf )

=

∫ Ex0

dEγfE1(Eγ)E

3γ

ρ(Ex, Ji, πi)

Ji+1∑Jf=|Ji−1|

ρ(Ex − Eγ , Jf , π̄i)

where the E1 operator requires a change of parity for thefinal state, here denoted as π̄i. The same selection rulesas given in footnote 6 have been applied, but we do notuse the approximation of Eq. (C.8), as many nuclei have ainitial spin Ji of 0 or 1/2. Further, we assume parity equili-bration of the level density, i.e. ρ(Ex, J,+) ≈ ρ(Ex, J,−),and express the level density ρ(Ex, J, π) through the spin-distribution g(Ex, J), Eq. (C.7),

〈Γγ,E1(Ex, Ji, πi)〉 =1

ρ(Ex, Ji, πi)(D.7)

×∫ Ex

0

dEγ

[fE1(Eγ)E

3γρ(Ex − Eγ)

Ji+1∑Jf=|Ji−1|

g(Ex − Eγ , Jf )2

].

At this point, to obtain the expression for M1 radiation,one only needs to exchange the E1 strength function fE1by the M1 strength function fM1. Using Eq. (D.2), wefind

〈Γγ(Ex, Ji, πi)〉 =1

2ρ(Ex, Ji, πi)(D.8)

×∫ Ex

0

dEγ

[[fE1(Eγ) + fM1(Eγ)

]E3γρ(Ex − Eγ)

×Ji+1∑

Jf=|Ji−1|g(Ex − Eγ , Jf )

].

This is equivalent to Eq. (2.11) in Ref. [89] under the as-sumptions listed above. One can determine 〈Γγ(Ex, Ji, πi)〉from neutron capture experiments, where usually slow neu-trons are used, thus Ex ≈ Sn. This is shown e.g. in Ref.[90], but we will repeat the derivation to get a comprehen-sive picture. The intrinsic spin of the neutron is 1/2, sowith capture of order ` on a target (denoted with the sub-script t) the entry states in the residual nucleus have the

possible spins Ji = Jt±1/2±` and parity πi = πt(−1)`. Fors-wave capture on a target nucleus with Jt = 0, there isonly one possible Ji, and we can directly compare the ex-perimental measurements to the calculations using (D.8).For Jt > 0, often only the average over all resonance ofdifferent Ji is reported. Using the level density ρ(Sn, Ji)of the accessible Ji’s, we find

〈Γγ`(Sn)〉 =∑Jiρ(Sn, Ji)〈Γγ(Sn, Ji, πi)∑

Jiρ(Sn, Ji)

(D.9)

=Dl2

∫ Sn0

dEγ

[f(Eγ)E

3γρ(Sn − Eγ)

×∑Ji

Ji+1∑Jf=|Ji−1|

g(Sn − Eγ , Jf )],

where we substituted∑Jiρ(Sn, Ji) by the average neu-

tron resonance spacing Dl and used the assumption of thedominance of the dipole decay to rewrite the sum of theaverage M1 and E1 strengths as f(Eγ). Note that tran-sitions for the highest γ-ray energies go to discrete states,not a quasi(-continuum). Thus, this integral is an approx-imation, and more precise calculations could distinguishbetween a sum over transitions to discrete states and theintegral for the (quasi-)continuum region.

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