Independent Normalization for γ -ray Strength Functions: The Shape Method
M. Wiedeking, M. Guttormsen, A.C. Larsen, F. Zeiser, A. Görgen, S. N. Liddick, D. Mücher, S. Siem, A. Spyrou
IIndependent Normalization for γ -ray Strength Functions: The Shape Method M. Wiedeking,
1, 2, ∗ M. Guttormsen, A.C. Larsen, F. Zeiser, A. G¨orgen, S. N. Liddick,
4, 5
D. M¨ucher,
6, 7
S. Siem, and A. Spyrou
4, 8, 9 Department of Subatomic Physics, iThemba LABS,P.O. Box 722, Somerset West 7129, South Africa School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa Department of Physics, University of Oslo, NO-0316 Oslo, Norway National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA Joint Institute for Nuclear Astrophysics, Michigan State University, East Lansing, Michigan 48824, USA (Dated: October 30, 2020)The Shape method, a novel approach to obtain the functional form of the γ -ray strength function( γ SF) in the absence of neutron resonance spacing data, is introduced. When used in connectionwith the Oslo method the slope of the Nuclear Level Density (NLD) is obtained simultaneously. Thefoundation of the Shape method lies in the primary γ -ray transitions which preserve informationon the functional form of the γ SF. The Shape method has been applied to Fe, Zr,
Dy, and
Pu, which are representative cases for the variety of situations encountered in typical NLD and γ SF studies. The comparisons of results from the Shape method to those from the Oslo methoddemonstrate that the functional form of the γ SF is retained regardless of nuclear structure detailsor J π values of the states fed by the primary transitions. I. Introduction
The number of nuclear levels per energy interval, thenuclear level density (NLD), and the γ -ray strength func-tion ( γ SF), which is a measure of the average reduced γ -ray decay probability, have received significant exper-imental and theoretical attention over the last decade.The necessity for reliable γ SF data has compelled theInternational Atomic Energy Agency to establish a ded-icated γ SF database together with recommendations [1].The demand for γ SFs and NLDs is driven in part due totheir relevance to astrophysical nucleosynthesis via cap-ture processes [2–5]. Recent experimental results haveclearly demonstrated that capture cross sections can bereliably obtained using NLDs and γ SFs as input intoreaction models [6–9], which are based on the Hauser-Feshbach approach [10].Several experimental methods exist [1] to extract γ SFsfrom experimental data, and of those the Oslo method[11] has been extensively used. The advantage of theOslo method lies in its ability to simultaneously extractthe γ SF and NLD from particle- γ coincident data. TheNLD and γ SF are traditionally normalized by three ex-ternal parameters: i) the NLD is normalized to the leveldensities of discrete states at low excitation energies, ii)the NLD at the neutron separation energy ( S n ) is con-strained to the s-wave neutron resonance spacing ( D ),and iii) the absolute value of the γ SF is determined fromthe average total radiative width of s-wave resonances ∗ [email protected] ( (cid:104) Γ γ (cid:105) ). The functional form of the NLD is linked tothat of the γ SF and can be fully constrained by normal-ization i) and ii) above. The γ SFs extracted with theOslo method have been shown to be reproduced usingthe alternative χ and Ratio methods, which do not relyon external models or normalization [12–14].Difficulties in normalizing NLD and γ SF data fromthe Oslo method emerge for nuclei without available D and/or (cid:104) Γ γ (cid:105) values. This is the case for many nuclei A when A − D and (cid:104) Γ γ (cid:105) data present challenges for the normaliza-tion of NLDs and γ SFs. In the absence of normalizationdata, no coherent prescription is currently available ascase-specific approaches [7, 8, 15–17] do not appear to beconsistently applicable. Even in cases where D is known,the normalization procedure introduces a model depen-dence, which can lead to large uncertainties [1]. A reliableapproach is highly desirable, especially since the requireddata needs driven by nucleosynthesis studies primarilyinvolve nuclei for which direct measurements of capturecross sections as well as D and (cid:104) Γ γ (cid:105) values are notpossible. Experimentally, γ SF and NLD data for nucleiaway from the line of stability are readily reachable how-ever, in particular with recent advances in extending theOslo method to previously inaccessible regions throughthe β -Oslo [7, 15, 18] and inverse-Oslo [19] methods.In this paper, the Shape method is introduced, whichis a novel and mostly model independent approach todetermine the slope of NLDs and γ SFs extracted withthe Oslo method in the absence of measured D values.We have also applied the Shape method to β -decay data a r X i v : . [ phy s i c s . d a t a - a n ] O c t on Ge and Kr to explore the extraction of model-independent NLDs away from stability [20]. In sectionII the Oslo method and the normalization for NLDs and γ SFs are reviewed. Section III presents the concepts anddetails of the Shape method, which allows for the nor-malization of NLDs and γ SFs. Section IV focuses on theShape method analysis and results on Fe, Zr,
Dyand
Pu. The discussion of results together with rec-ommendations on the use and applicability of the Shapemethod is provided in section V. Summarizing remarksare made in section VI.
II. Review of the Oslo Method and Normalizations
Fermi (cid:48) s golden rule [21] states that the decay rate λ if from an initial ( i ) state to a distribution of final ( f ) statesis given by a product of the density of final states ρ f andthe transition probability |(cid:104) f | H (cid:48) | i (cid:105)| : λ if = 2 π (cid:126) | (cid:104) f | H (cid:48) | i (cid:105) | ρ f , (1)where H (cid:48) is the electromagnetic transition operator.The Oslo method [11] extracts the γ SF and NLD simul-taneously through the following procedure: States in thequasi-continuum (below the particle threshold) are typi-cally populated with charged-particle direct and scatter-ing reactions or following β decay. The γ -ray spectrumis unfolded with the detector response function using aniterative subtraction technique [22]. From the unfoldedspectra, and with the assumption that the residual nu-cleus reaches a compound state, the primary γ -ray spec-trum is obtained through the first-generation method[23]. The first-generation matrix P ( E i , E γ ) is propor-tional to the γ -ray decay probability and can be fac-torized according to the expression that is derived fromFermi (cid:48) s golden rule (details are found in App. C of Ref.[24]) P ( E γ , E i ) ∝ ρ ( E f ) T ( E γ ) , (2)where ρ ( E f ) is the nuclear level density and T ( E γ ) is thetransmission coefficient, which is independent of excita-tion energy ( E i ) and hence nuclear temperature. Thisfollows from the generalized Brink-Axel hypothesis [25],which states that collective excitation modes built on ex-cited states have the same properties as those built onthe ground state. The hypothesis has been validated inthe quasi-continuum with the Oslo method [26]. Thetheoretical matrix P th ( E γ , E i ) is given by [11] P th ( E γ , E i ) = ρ ( E f ) T ( E γ ) (cid:80) E γ ρ ( E f ) T ( E γ ) . (3)The ρ ( E f ) and T ( E γ ) can be simultaneously extractedby performing a χ minimization between the theoretical P th ( E γ , E i ) and experimental P ( E γ , E i ) first-generationmatrices [11].From Eq. (3) an infinite number of solutions are ob-tained, and the physical solution is found by normalizing T ( E γ ) and ρ ( E f ) to experimental data [11] with˜ ρ ( E f ) = Aρ ( E f ) e αE f , (4)and ˜ T ( E γ ) = B T ( E γ ) e αE γ , (5)where A and B are constants and α is the common slope .The slope α and constant A are determined by the NLDof the known discrete states at lower excitation energiesand the total NLD at S n . The functional form of ρ ( E f )and T ( E γ ) is defined from the χ fit to the primary γ -raymatrix P ( E γ , E i ). For a detailed discussion and imple-mentation of the Oslo method, see Ref. [24].In this work, data from Fe [27], Zr [28],
Dy[29], and
Pu [30] have been reanalysed with the Oslomethod using an intrinsic spin-distribution for the abso-lute normalization at S n . The γ SFs of those nuclei maytherefore deviate slightly from results presented in pre-vious publications. The form of the spin-distribution isassumed to follow [31] g ( E, J ) (cid:39) J + 12 σ ( E ) exp (cid:2) − ( J + 1 / / σ ( E ) (cid:3) , (6)where E is the excitation energy, J the spin, and the spincutoff parameter σ ( E ) is assumed to have the functionalform σ ( E ) = σ d + E − E d S n − E d (cid:2) σ ( S n ) − σ d (cid:3) , (7)determined by two excitation energies. At the lower ex-citation energy E = E d , we determine the spin cutoffparameter σ d from known discrete levels. The secondpoint at E = S n is estimated assuming a rigid momentof inertia [32, 33] σ ( S n ) = 0 . A / √ aU n a , (8)where A is the mass number, a is the NLD parameter, U n = S n − E is the intrinsic excitation energy, and E is the energy-shift parameter.At S n , normalization is achieved from NLDs calculatedwith [11] This is an additional slope transforming ρ ( E f ) and T ( E γ ) inthe same way as for ˜ ρ ( E f ) and ˜ T ( E γ ). Note however, that theslopes of ρ ( E f ) and T ( E γ ), and ˜ ρ ( E f ) and ˜ T ( E γ ) are in generaldifferent. ρ ( S n ) = 2 σ ( S n ) D J + 1) e (cid:104) − ( J +1)22 σ Sn ) (cid:105) + Je (cid:104) − J σ Sn ) (cid:105) . (9)The experimental D value is obtained from (cid:96) = 0 (s-wave) neutron resonance spacing data which are typi-cally retrieved from Refs. [34, 35] and J is the initialspin of the target nucleus. Generally, NLDs can only beextracted to excitation energies well below S n with theOslo method. The absolute normalization at S n , whichsensitively depends on the spin distribution, is achievedby extrapolating the NLDs using a variety of level densitymodels, such as the back-shifted Fermi-gas [36], the con-stant temperature [37], or the Hartree-Fock-Bogoliubov-plus-combinatorial [38] models.The absolute normalization parameter B in Eq. 5 isobtained by constraining the experimental data to (cid:104) Γ γ (cid:105) for s-wave resonances by [24, 39] (cid:104) Γ γ ( S n ) (cid:105) = 12 πρ ( S n , J t ± / , π t ) × (cid:88) J f (cid:90) S n B T ( E γ ) ρ ( S n − E γ , J f ) dE γ , (10)where π t is the parity of the target nucleus in the (n, γ )reaction, J f and J t are the spins of the levels in the finaland target nucleus, respectively.The essential parameters used here for the extractionof the NLDs and γ SFs are listed in Table I. More detailson the extraction of NLDs and γ SFs for Fe, Zr,
Dy,and
Pu are discussed in Refs. [27–30].The relationship between T ( E γ ) and the γ SF( f XL ( E γ )) with XL being the type and multipolarityof the radiation, respectively, is [34] T XL ( E γ ) = 2 πE γ L +1 f XL ( E γ ) . (11)With the assumption that statistical γ -ray decay is dom-inated by dipole transitions, the total γ SF ( f ( E γ )) be-comes f ( E γ ) = f E ( E γ ) + f M ( E γ ) = T ( E γ )2 πE γ . (12)The values of D and (cid:104) Γ γ (cid:105) from s-wave resonance andto a limited extent D and (cid:104) Γ γ (cid:105) values from p-wave res-onance measurements are generally available for nucleiwhich are populated through (n, γ ) reactions on stable A similar treatment as for D can be applied to p-wave neutronresonance spacing data ( D ) and if available may be used toprovide additional constraints. targets. For the majority of nuclei the information re-quired by the Oslo method to determine A , B , and α hasnot been measured mostly due to the unavailability oftargets. This led to many non-standardized approachesto estimate the values D and (cid:104) Γ γ (cid:105) [7, 8, 15–17].The development of a method with no or only verylimited model dependencies, which can be systematicallyapplied to nuclei, is of utmost importance to obtain thenormalization when D and (cid:104) Γ γ (cid:105) values are not avail-able. A new method, the Shape method, will now bedescribed, which provides a prescription for the normal-ization of the slope of the NLD and γ SF in the absence of D . Software for the Oslo and Shape ( diablo.c ) meth-ods are available from Refs. [24, 41]. III. The Shape Method
In this section, the Shape method, which is a techniqueto obtain the slope of γ SF in the absence of measuredvalues of resonance spacing, is presented. The methodutilizes concepts from γ SF measurements using the av-erage resonance proton capture approach and from theRatio and χ methods using particle- γ - γ coincident data.These approaches are briefly summarized before we con-tinue with a detailed description of the Shape method. A. Average Resonance Proton Capture
Experimental data from ( p , γ ) reactions have been usedto deduce the γ SFs for several 45 < A <
91 nuclei forwhich the proton separation energy ( S p ) is located below S n [1]. The methodology is similar to the neutron averageresonance capture approach [42] where several resonancesare populated and combined in specific excitation-energyranges. The use of high-resolution detectors allows forthe identification of individual primary γ -ray transitionsto low-lying levels, see for example Refs. [43, 44]. Therelative intensities of primary transitions (corrected by E γ ), which originate from a given excitation energy re-gion and decay to low-lying levels with the same spin andparity, preserve the shape and hence the energy depen-dence of the γ SF. The proton beam energies, togetherwith the target thicknesses, provide an unambiguous as-signment of specific excitation energies. Data of primarytransitions to low-lying states of different spins and par-ities ( J π ) are normalized by weighting the different con-tributions through the Hauser-Feshbach formalism. Re-gardless this normalization, the energy dependence of the γ SF remains completely independent of any model input.
B. Ratio and χ Methods
The Ratio method [12] is a model-independent ap-proach to obtain the energy dependence of the γ SF fromcorrelated particle- γ - γ events following direct reactions. TABLE I. Parameters used for the extraction of NLDs and γ SFs (see text for details).
Nucleus S n D a c E c E d σ d σ ( S n ) ρ ( S n ) T CT (cid:104) Γ γ (cid:105) [MeV] [eV] [MeV − ] [MeV] [MeV] [MeV − ] [MeV] [meV] Fe 11.197 - 6.196 0.94 2.70 2.5 4.05 2870(680) † † Zr 8.635 514(15) a Dy 7.658 6.8(6) b × Pu 6.534 2.20(9) b × † Estimated from systematics corresponding to norm-2 in Ref. [40]. a Value from [35]. b Value from [34]. c Values from [32, 33].The γ - γ coincidence is between the primary γ -ray transi-tion, originating from the region of the quasi-continuumpopulated in the reaction, and the transition from low-lying discrete states, which are fed by the primary γ rays.When a discrete transition from a low-lying state is de-tected in coincidence with a charged particle, additionalstringent requirements are applied to the primary γ ray,so that the energy sum of the discrete and primary tran-sitions is equal to the excitation energy within the en-ergy resolutions of the detectors. Any particle- γ - γ eventsatisfying these conditions provides an unambiguous de-termination of the origin and destination of the observedprimary transition. As long as the primary γ rays feeddiscrete states of the same J π the shape of the γ SF re-mains independent of model input by analogy with the(p, γ ) average resonance proton capture method. The ra-tio R of intensities N for two different primary γ -rayenergies from the same initial excitation energy E i todiscrete low-lying levels of same J π at energies E l and E l is R = f ( E i − E l ) f ( E i − E l ) = N l ( E i )( E i − E l ) N l ( E i )( E i − E l ) . (13)When the ratios from different excitation energies arecompared, information on the energy dependence of the γ SF is obtained as demonstrated from (d,p γγ ) [12],(p,p’ γγ ) [14], ( (cid:126)γ, γγ ) [45], and (p, γ ) [46] reactions.Data of primary γ -ray intensities from an excitationenergy range to different discrete levels of the same J π and corrected for E γ , can also be fitted with a χ min-imization procedure [12–14]. The set of data from dif-ferent initial excitation energies are independent of eachother and following the χ minimization, which combinesthe sets from different excitation energy bins, yields in-formation on the shape of the γ SF.
C. Shape Method
In the previous descriptions discrete γ -ray lineswere studied with high-resolution germanium detectors.When the total γ SF extending across larger excitationand γ -ray energy ranges is to be measured, the Oslo method with high-efficiency detectors is regularly used.In the following, we will extend the previous techniquesand replace the identification of γ -ray lines from discretelevels l j with diagonals D j in a particle- γ matrix.The diagonals D j are directly related to the first-generation (or primary) P ( E γ , E i ) matrix provided bythe Oslo method. Figure 1 illustrates the concepts ofdiagonals and symbols used where one may define a finalexcitation energy E f fed from an initial excitation en-ergy E i by a γ transition with energy E γ . This is givenby E i ( E γ ) = E γ + E f with E f fixed and the diagonals D j with different E f are parallel to each other as schemati-cally shown in Fig.1. Here, the direct γ -ray decay from E i to the ground state is simply given by E i ( E γ ) = E γ (within the resolutions of the detectors). The diagonalsmay appear in three variants containing ( i ) one final statewith given J π , ( ii ) two or more specific final states, orin case of high level density, ( iii ) a large number of fi-nal states (typically >
20) with a corresponding average E f and J π . The number of counts along a D j relates tothe γ SF for a given E γ originating from E i . The inten-sities (counts) given by the content of the pixel ( E γ , E i )for two diagonals are exploited to obtain a pair of datapoints which are proportional to the γ SF.In the following, we assume a symmetric parity dis-tribution with the spin distribution g ( E i , J i ) of Eq. (6).Furthermore, we assume the population of a typical stateat excitation E i and spin J i is given by the cross section σ ( E i , J i ). The number of counts in a diagonal D j at( E γ , E i ) with one or more final J π states included, canthen be expressed as a sum of products N D ∝ (cid:88) [ J f ] J i = J f +1 (cid:88) J i = J f − σ ( E i , J i ) g ( E i , J i ) G ( E i , E γ , J i , J f ) , (14)where we define [ J f ] as the spins of the final levels withinthe diagonal, e.g. [ J f ] = [1 − , + , + , − ] includes thesumming of four terms. The second sum is restrictedto the available J π populated by dipole transitions con- The total γ -ray matrix (all γ rays in a cascade) may be utilized,as long as it is certain that the diagonals contain only primarytransitions. D2 D1E f = 1 MeV E f = 0 MeV E g E i FIG. 1. (Color online) Illustration of diagonals (blue) D and D selecting specific final states in the P ( E γ , E i ) matrix. Hor-izontal bars (yellow) indicate three initial excitation energies E i . The number of counts at the crossing points between adiagonal and a bar ( E γ , E i ) gives the intensity of the γ tran-sitions from E i to E i − E γ , here symbolized with filled circles,squares, and triangles. With intensities from two diagonals atthe same E i , a pair of internally normalized γ SF data pointscan be established. necting initial and final states, which generally includesthree initial spins. However, in the case of J f = 0, onlythe J i = 1 spin is included and for J f = 1 /
2, only the J i = 1 / J i = 3 / G in Eq. (14) is proportional to the γ -decay width given by G ( E i , E γ , J i , J f ) ∝ (cid:90) E γ +∆ / E γ − ∆ / T ( E i , E (cid:48) γ , J i , J f ) δ ( E i − E (cid:48) γ , J f ) dE (cid:48) γ , (15)where ∆ is the energy width of the diagonal which in-cludes the specific final level J f at E f = E i − E γ . The δ function assures that one specific level is counted giv-ing (cid:82) δ dE (cid:48) γ = 1. With the assumption that the trans-mission coefficient is almost constant within this energybin, it can be placed outside the integral with a value of T ( E i , E γ , J i , J f ).According to the generalized Brink-Axel hypothesis,the transmission coefficient T ( E i , E γ , J i , J f ) is assumedto be independent of spin and excitation energy. Thus,we replace the expression for the transmission coefficientby T ( E γ ), i.e. a function only dependent of E γ . Further-more, if we assume the dominance of dipole transitions inthe quasi-continuum region, the transmission coefficientcan be replaced by the γ SF through T ( E γ ) = 2 πf ( E γ ) E γ from Eq. (12).With the considerations above, Eq. (14) can be written as N D ∝ f ( E γ ) E γ (cid:88) [ J f ] J i = J f +1 (cid:88) J i = J f − σ ( E i , J i ) g ( E i , J i ) . (16)In the following we will assume that the probability topopulate a certain initial state with spin J i at a given E i is approximately independent of spin, i.e. σ ( E i , J i ) ≈ σ ( E i , J (cid:48) i ).The Shape method applies for the same E i but for twodifferent diagonals D and D , see Fig. 1. We choosediagonal D to represent a lower final excitation energy E f and D a higher final excitation energy E f . Atthe initial excitation energy E i , the γ -ray energies are E γ = E i − E f and E γ = E i − E f for diagonals D and D , respectively.The strength functions at E γ and E γ are determinedby the number of counts at the diagonals D and D forthe same initial excitation energy E i , using Eq. (16) f ( E γ ) ∝ N D E γ (cid:80) [ J f ] (cid:80) J i = J f +1 J i = J f − g ( E i , J i ) f ( E γ ) ∝ N D E γ (cid:80) [ J f ] (cid:80) J i = J f +1 J i = J f − g ( E i , J i ) . (17)In synergy with the methods introduced above, such apair of γ SF data points is internally normalized and wecan determine a γ SF data-point pair for each E i . Thedouble sum can be omitted if the two diagonals includeone final level each of the same J π . However, such di-agonals are often difficult to identify in the data, and itis more common to observe different spins for two diago-nals, such as the 0 + ground state and the first-exited 2 + state in even-even nuclei.Figure 2 illustrates a sewing technique that allows toconnect pairs of γ SF data points and is the final step ofthe Shape method to obtain the functional form of the γ SF. In this example, we show three different pairs, eachfrom a different E i , marked by filled circle, square andtriangle data points. The second and third γ SF pairs arescaled as explained in the figure caption. In detail, this isaccomplished by finding the average γ -ray energy E γ ave (location of arrow) in between the lowest and highest γ SFdata points of the two pairs under study. Then we use alogarithmic interpolation of the γ SF data points for eachpair to E γ ave . The resulting sewed γ SF is represented bythe black line to guide the eye in panel (c) and exhibitsthe shape of the γ SF.
IV. Shape Method Analysis and Results
In the following, when referring to discrete final lev-els within the diagonals, we always refer to levels inthe data base from the National Nuclear Data Center(NNDC) [47]. For each application of the Shape Methodwe use a first-generation matrix with ≈ −
40 keV/ch g -ray energy E g g - r a y s t r e n g t h f un c t i o n f ( a . u . ) (a)(b)(c) FIG. 2. Illustration of the sewing technique for three γ SFpairs (filled circles, squares and triangles) with each pair con-nected by dashed lines in (a). The second pair of data points(filled squares) is scaled by a factor to match the first pair ofdata points at a location indicated by the arrow (filled cir-cles) (a). Then the third pair of data points (filled triangles)is scaled to match the previously corrected data pair (filledsquares) at the location of the arrow (b). Finally, the resultingsewed γ SF is presented in (c) (solid black line). on both axes from which the number of counts are de-termined through integration. These are then furthercompressed into bins of ≈
120 keV/ch unless otherwisenoted. Detailed discussions on the comparisons of theresults from the Shape and Oslo methods are deferred toSec. V.
A. Diagonals with the same final J π : Fe We utilize data from the Fe( p, p (cid:48) γ ) Fe reaction pre-viously presented in Refs. [27, 40], where the γ rayswere measured with six large-volume LaBr (Ce) detec-tors from the HECTOR + array [48] and the charged par-ticles with the SiRi silicon telescope [49]. Figure 3a showsthe resulting P ( E γ , E i ) matrix of Fe. Gates were seton the diagonals and correspond to the direct decays tothe 2 +1 (diagonal D ) and 2 +2 (diagonal D ) levels at 847keV and 2658 keV in Fe, respectively. As the spins andparities for the two final levels are equivalent, it is rea-sonable to assume that the initial level density ρ ( E i ) andthe population-depopulation factor σ ( E i , J i ) g ( E i , J i ) ofthe initial levels that feed the final states in the diago-nals are also the same. Therefore, the number of countsin the diagonals for a given E i only needs to be correctedby the E γ factor. Following the sewing steps outlinedabove for the pairs of intensities for each E i , the shape ofthe γ SF is obtained and compared to the results of the Oslo method in Fig. 3b.Due to the lack of neutron-resonance spacing datafor Fe, as Fe is unstable, previous works have re-lied on systematics to obtain the slope of the NLD and γ SF [27, 40]. Comparing the previous results with thoseof the new Shape method, we can conclude that thetwo normalizations previously used are indeed reason-able. However, as there is only a ∼
30% relative changein the estimated NLD at S n ( ρ ( S n ) = 2.18(59) MeV − and 2.87(68) MeV − ) between the two normalizations,we are not in a position to confirm which normalizationis correct. If there was a more pronounced discrepancyin slope between the different normalizations, the presentmethod may enable a discrimination between the inputspin-distribution models. Although the systematics usedin Fe is appropriate there is no compelling reason toassume that systematic approaches can be extended toall nuclei. Hence, if no reliable systematics can be made,such as for nuclei far away from stability, the presentmethod, which is based on a sound foundation, clearlyprovides a significant constraint on the slope of the NLDand γ SF. The low and high-energy discrepancies observedin Fig. 3b are further explored in Sec. V.
B. Several Diagonals with different final J π combinations: Zr Data from the (p,p’) reaction populating Zr [28] wereused with the γ rays detected in the NaI(Tl) CACTUS ar-ray [50] and the charged particles in SiRi. With N = 52, Zr is close to the magic N = 50 shell closure and ischaracterized by few low-lying levels. With the presentexperimental resolution it is possible to identify four diag-onals. With the six combinations D D , D D , D D , D D , D D , and D D one can investigate the con-sistency between the various γ SFs from the Shape andOslo-method results.Figure 4a shows the primary matrix with the diagonals D j which include the following discrete states: D : 0 + (0 keV) D : 2 + (934 keV) D : 0 + (1383 keV) and 4 + (1495 keV) D : 3 − (2340 keV), 4 + (2398 keV), and 5 − (2486keV).The lower part of the matrix shows that many non-statistical γ -ray transitions connect discrete levels andit is important to point out that these should not betaken into account when extracting the average γ SF for Zr. Thus, the results for the Oslo method in Fig. 4bwas extracted for E i > . g E -ray energy g ( M e V ) i E E xc i t a t i on ene r g y C oun t s C oun t s D D Fe ) g Fe(p,p' (a) g E -ray energy g - - ) - ) ( M e V g E ( f -r a y s t r eng t h f un c t i on g Oslo method rays feeding D g rays feeding D g (b) FIG. 3. (Color online) (a) The first-generation matrix P ( E γ , E i ) of Fe showing the cuts on the diagonals decaying to the 2 +1 level ( D ) at E f = 847 keV and the 2 +2 level ( D ) at E f = 2658 keV. (b) The resulting γ SF from the Shape method (filledand open blue triangles) compared to the Oslo-method results (solid black squares) [27, 40]. Note that the bin width is 248keV/ch in this case due to Fe being a relatively light nucleus with a low level density. g E -ray energy g ( M e V ) i E E xc i t a t i on ene r g y C oun t s C oun t s D D D D Zr ) g Zr(p,p' (a) g E -ray energy g - - - ) - ) ( M e V g E ( f -r a y s t r eng t h f un c t i on g Oslo method rays feeding lower D g rays feeding upper D g (b) D D D D D D D D D D D D FIG. 4. (Color online) (a) The first-generation matrix P ( E γ , E i ) of Zr showing the four diagonals described in the text. (b)The resulting γ SFs from the Shape method (filled and open triangles in blue and light blue) compared to the Oslo-methodresults (black squares). The individual Shape method results are shifted in the plot in order to visualize the results from thevarious combinations of diagonals. properties there should be enough initial states withinthe energy bin at E i that feed the levels contained bythe diagonals. For Zr we obtain erratic fluctuations for E γ < γ SFs from theShape method are all in rather good agreement withthe functional form between each other and the one ob-tained with the Oslo method. Since the combination of diagonals represent a variety of final J π values, yet theyprovide consistent functional forms, the spin distribution g ( E, J ) applied in Eq. (6) with spin cutoff parameters ofTable I is supported.
C. Diagonals including Ground andTwo-Quasiparticle Bands: Dy For rare earth nuclei the level density becomes highenough that it is difficult to identify final levels in the P ( E γ , E ) matrix within the experimental resolutions.However, the known levels of Dy group into the groundband between 0 − . Dy a feasible case for applying the Shape method tothe
Dy( He, He’) experimental data, measured withthe CACTUS and SiRi arrays, from Refs. [29, 51, 52].Furthermore, there are two interesting features in theprevious findings of the γ SF: ( i ) a scissors resonance at E γ = 2 . ii ) it has been speculated if an enhance-ment exists around E γ = 6 − E ) - ( E ) ( M e V r Le v e l den s i t y Oslo method Known levels CT model from neutron res. data r Dy D D FIG. 5. Level densities of
Dy [52]. The solid line rep-resents the NLD of known levels. The filled square symbolsshow the results of the Oslo method. The data points are con-nected to the NLD at S n (open square) through extrapolationwith the constant temperature (CT) model. ately recognise the diagonals corresponding to the groundand two-quasiparticle bands by inspecting the distribu-tion of known levels. Here, diagonal D includes the 0 + ,2 + , 4 + and 6 + levels of the ground state band in the ex-citation region of 0 − . D includes 14levels in the excitation region of 0 . − .
39 MeV, all withknown J π [47]. Figure 6b shows the γ SF extracted withthe Oslo method [29] together with the Shape methodresults.It is interesting to note that the scissors resonance isdirectly visible from Fig. 6a as a yellow-shaded region for E i > E γ ∼ − E γ ∼ − S n with an apparent deviation in slope at E γ ∼ . E γ ∼ − E γ > . D. Diagonals with many final levels of different J π : Pu The
Pu isotope was populated in the (d, p) reactionwith a beam energy of 12 MeV and the γ rays detectedwith the CACTUS and charged particles with the SiRiarrays. The excitation energy range analyzed here wasrestricted to E i < . S n = 6 .
534 MeV. Further details of the experimentalset-up and considerations are given by Ref. [30] and allresults presented here are based on a reanalysis of thedata.The low-spin transfer of this sub-Coulomb barrier re-action is responsible for the population of only a frac-tion of the total intrinsic levels. An iterative procedurewas developed [30] that aims to correct for the bias in-troduced in the Oslo method. The populated J π distri-butions were estimated by the Green’s function transferformalism and applied in γ -decay simulations to obtainconsistent results [30, 53, 54]. In the following, we ex-plore the possibility to apply the Shape method, eventhough the calculated J π distribution may not fulfill theassumptions on σ ( E i , J i ) specified in Sec. III. If the Shapemethod can be used to reliably extract the slope of the γ SF, it would be significantly easier to apply it than theiterative procedure proposed in Ref. [30].A reduced spin population may be a challenge for theOslo method since it is not clear what effect a vary-ing J π population σ ( E i , J i ) has on the first-generationmethod [30, 55, 56]. Nonetheless, we will now assumethat σ ( E i , J i ) does not significantly impact the overallresults of the first-generation matrix P ( E γ , E i ). To ac-count for the fact that high-spin levels are rarely popu-lated in the sub-Coulomb barrier reaction, the level den-sity ρ ( S n ) used in the decomposition of P ( E γ , E i ), seeEq. (2) and Eq. (4), has to be reduced by a factor r . Thisfactor is directly linked to the slope of the γ SF throughthe normalization Eqs. (4) and (5), such that it can bedetermined by a comparison of the γ SF from the Osloand the Shape methods.The key for an investigation with the Shape method is g E -ray energy g ( M e V ) i E E xc i t a t i on ene r g y C oun t s C oun t s D D Dy ) g He' He, Dy( (a) g E -ray energy g - - - ) - ) ( M e V g E ( f -r a y s t r eng t h f un c t i on g Oslo method-rays feeding D1 g -rays feeding D2 g (b) FIG. 6. (Color online) (a) The first-generation matrix P ( E γ , E i ) of Dy showing the two diagonals described in the text.(b) The resulting γ SF from the Shape method (filled and open blue triangles) compared to the Oslo-method results (blacksquares). to identify two diagonals in the P ( E γ , E i ) matrix whichinclude a known number of final levels with proper spinassignments. The Pu isotope is one of the best studiednuclei in this mass region with a complete level schemeup to ≈ J <
5, which can be used to define the diagonals. Thediagonal D includes the first 0 + , 2 + and 4 + levels withan average final excitation energy of E f = 62 keV. Thesecond diagonal D has nine levels between 0 . − . E f = 849 keV and an averagespin of 2.3 (cid:126) . Figure 8a shows the two diagonals chosenand the resulting γ SF pairs are presented in Fig. 8b. Theslope of the γ SF obtained with the Shape method is inagreement to the slope obtained with the Oslo methodwhen a reduction factor of ≈ . ρ ( S n ). Wealso display the γ SF if one assumes that all spins arepopulated in the reaction, which displays a significantlysteeper slope. The corresponding NLDs used to extractthe γ SFs are shown in Fig. 7.It is difficult to make rigorous conclusions on the re-duction factor r since diagonal D may have missing lev-els. In addition, there are uncertainties at the upperlimit of 1 MeV to determine which levels are includedwithin the experimental detection resolution. Thus, thecase of Pu is meant to highlight the possibilities thatmay exist if reliable diagonals can be defined with highexperimental resolution.It can be seen that the resulting γ SF is relatively flatbetween E γ ∼ − . σ ( E i , J i ), or whether there is a strongenough contribution of e.g. the scissors resonance be-tween 2 and 4 MeV that leads to an almost constant tailof the γ SF within the narrow E γ range considered. V. Discussion
The shapes of the γ SFs extracted with the Oslomethod are well reproduced with the Shape method,in particular for excitation energies for which the to-tal NLD of initial states is high. With reduced excita-tion energies discrete structures may become dominantand the concepts of γ SF and NLD are no longer appli-cable. This situation is apparent when inspecting the γ SF of Fe in Fig. 3b where the γ SF below E γ ∼ . E i ∼ . E i = 6 MeV has been measured to be ρ ∼
100 MeV − [57]. For Zr the Shape method hasbeen applied from E i = 4.5 MeV where ρ ∼
180 MeV − [28]. For the heavier nucleus Dy the level densityreaches ρ ∼
800 MeV − at E i = 3 MeV and for Pu ρ ∼ − at E i = 2 . Dy and
Pu allow for the Shape method to be ap-plied to low enough E i values to cover the range of thescissors resonance. It is important to emphasize thatcareful considerations have to be given to identify appro-priate E i regions for the Shape method to be applicable.Discrete states and/or structures may become dominantfeatures which lie outside the statistical regime. This isparticularly the case for light A nuclei or those which arelocated near closed shells. From our investigation, a min-imum of ρ ∼
100 MeV − appears to be appropriate, ormore specifically, one should have more than ≈
10 tran-sitions connecting the initial and final excitation energybins. It is nonetheless recommended that each nucleusis being investigated carefully to determine the lowestreliable E i and hence lowest γ -ray energy to be used.0 ) - ( E ) ( M e V r Le v e l den s i t y ) n (S r Oslo method ) n (S r · Oslo method 0.1 Known levels CT model Pu D D FIG. 7. Level densities of
Pu. The solid line shows theNLD of known levels. The open squares represent the re-sults of the Oslo method when ρ is normalized to the totallevel density at S n , while the filled squares show the resultswhen the reduced population of high-spin levels is taken intoaccount. The reduction factor 0 . γ SFs from the Oslo and the Shape method. Thedata points are extrapolated to the corresponding NLDs at S n with a constant temperature (CT) model (dashed lines).Note that the error bars are less than the size of the datapoints. At higher E i , the data points from the Shape methodfollow the functional form of the γ SFs from the Oslomethod rather well. At the highest E i , the Oslomethod may underestimate the γ SF due to reducedstatistics whereas the Shape method remains robust inthis regime. As demonstrated for the four nuclei underconsideration, it is in the region of higher γ -ray energieswhere the slope of the γ SF can be reliably obtained withthe Shape method and provides the necessary constraintsif alternative normalization procedures are not possibledue to the absence of neutron resonance data.Nuclei such as Fe, for which two low-lying discretestates of the same J π can be separated experimentally,represent the most fundamental application of the Shapemethod and can be treated with the fewest assumptionsand without any model input. In such cases, the NLDand cross section dependencies of primary transitionsfeeding the states are eliminated.The Shape method remains applicable even when thediscrete levels differ in J π or if the states cannot be re-solved experimentally. This is clearly demonstrated for For Fe the low statistics is due to the very few levels up to E i ≈ γ SF. For
Dy the matrix has low statistics at high energiesas indicated by the large uncertainties. Zr where six different combinations of final levels allyield strikingly similar functional forms of the γ SF. Thisillustrates the robustness of the applied spin distributionsand the assumption that the population cross-section isproportional to the spin distribution over the E i rangesconsidered for the extraction of γ SF below the particlethresholds.The results from
Dy further reveal that the inclu-sion of many final levels of widely varying J π values oreven distinctive nuclear structures still leads to an energydependence which is in agreement with that of the γ SFsfrom the Oslo method. The
Dy Oslo method resultsshow the presence of the scissors resonance. The sameinformation is retained in both diagonals and the reso-nance is reproduced by the Shape method. This may im-ply that this resonance is a collective mode obeying theBrink-Axel hypothesis. A suspected pygmy resonanceat E γ ∼ − Dy, while previous results were inconclusive[29], highlighting the complementary nature of the Shapemethod.
Pu represents an extreme case due to the reac-tion proceeding below the Coulomb barrier yielding avery limited spin-distribution. This requires ρ ( S n ) to bemodified through the Oslo method, which propagates tothe normalization of the γ SF, in order to reproduce theShape method results. It is important to note, once theappropriate corrections are performed that both methodsyield a similar energy dependence of the γ SFs despite theselectivity of the reaction. The reduced strength at E γ ∼ γ SFs, regardless if the γ SFs are built on different nuclear structures or J π statesof a given nucleus. This confirms the validity of the gen-eralized Brink-Axel hypothesis, supporting previous re-sults [26]. Another appealing aspect of the Shape methodis the fact that it can be applied to the same set of ex-perimental data as that used to extract the NLD and γ SF with the Oslo method. This is highly beneficialwhen the Shape method is used to specifically determinethe slope for the NLD and γ SF from the Oslo methodsince it avoids unnecessary additional systematic uncer-tainties which would arise when performing different ex-periments.
VI. Summary
It has long been a challenging endeavour to estimatethe slope of the γ SF in the absence of neutron resonancedata which is compounded by the fact that no standard-ized approach exists which is applicable to all nuclei. TheShape method provides a solution to the γ SF normaliza-tion conundrum when D values are not available. It pro-vides a standardized approach to determine the slope of1 g E -ray energy g ( M e V ) i E E xc i t a t i on ene r g y C oun t s C oun t s D D Pu ) g Pu(d,p (a) g E -ray energy g - - - ) - ) ( M e V g E ( f -r a y s t r eng t h f un c t i on g ) n (S r Oslo method ) n (S r · Oslo method 0.1 -rays feeding D1 g -rays feeding D2 g (b) 3 · FIG. 8. (Color online) (a) The first-generation matrix P ( E γ , E i ) of Pu showing the two diagonals described in the text. (b)The γ SF obtained with the Shape method (filled and open blue triangles) compared to the Oslo-method results with a reducedspin range (solid black squares). The results with a full spin distribution are shown as open black squares and multiplied by afactor of 3 to facilitate readability of the figure. the γ SF and NLD (if extracted simultaneously throughthe Oslo method), which is not only universally appli-cable but will also provide consistency for analyses andresults.The Shape method makes use of concepts from the Av-erage Resonance Proton Capture, Ratio, and χ methodsand is based on the unambiguous experimental identi-fication of the origin and destination of primary γ -raytransitions. Through their intensities, pairs of primarytransitions retain the information on the functional formof the γ SF.The Shape method has been applied to four nucleiwhich are representative of the various situations encoun-tered: i) low-mass Fe, ii) Zr located in the vicinityof shell closures, iii)
Dy with scissors and pygmy reso-nances, and iv) high- Z nucleus Pu where the reactionproceeds below the Coulomb barrier. These four nucleifurther represent a variety of J π combinations for low-lying states which are fed by the primary transitions.In Fe, the primary transitions feed two well-separated and experimentally-resolved states of the same J π , while in Zr some of the low-lying states cannot beresolved and are of different J π . For Dy the low-lyingstates can only be identified through clusters of specificnuclear structures in the form of the ground and two-quasiparticle bands. The
Pu case has an even largernumber of final states which cannot be resolved experi-mentally. Regardless of the intricacies and details of theindividual nuclei considered, the Shape method extractsfunctional forms of γ SFs which are consistent with thosefrom the Oslo method. This highlights the robustness ofthe method and, where applicable, the appropriateness of the assumptions made regarding the spin distributions.While the Shape method provides a universal prescrip-tion to determine the slope of the γ SF (and for the NLDin the case of the Oslo method) in the absence of experi-mentally measured neutron resonance spacing it does notprovide the absolute values of the γ SFs when neutron res-onance widths are not available. Further work is highlydesirable to explore alternate approaches to determinethe absolute values of γ SFs.Complementary to this work, we have also applied theShape method to Ge and Kr for the extraction ofmodel-independent nuclear level densities away from sta-bility [20].
Acknowledgments
This work is based on the research supported in part bythe National Research Foundation of South Africa (GrantNumber: 118846), by the Research Council of Nor-way (Grant Number: 263030), and the National ScienceFoundation (Grant Number: PHY 1913554). A. C. L.acknowledges funding of this research by the EuropeanResearch Council through ERC-STG-2014 under grantagreement no. 637686, support from the “ChETEC”COST Action (CA16117), COST (European Coopera-tion in Science and Technology), and from JINA-CEEthrough the National Science Foundation under GrantNo. PHY-1430152 (JINA Center for the Evolution ofthe Elements).2 [1] S. Goriely, P. Dimitriou, M. Wiedeking, T. Belgya, R.Firestone, J. Kopecky et al. , Eur. Phys. J.
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