Is the generalized Brink-Axel hypothesis valid?

0.73 MB·

by  M. Guttormsen

Is the generalized Brink-Axel hypothesis valid?
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M. Guttormsen∗a, A. C. Larsena, A. Görgena, T. Renstrøma, S. Siema, T. G. Tornyia,b
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2 and G. M. Tvetena
n aDepartmentofPhysics,UniversityofOslo,Norway
a bDepartmentofNuclearPhysics,AustralianNationalUniversity,Canberra,Australia
J
E-mail: [email protected]
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] Experimentalresultsofthe237Np(d,pγ)238Npreactionarepresented,whichverifiesthegeneral-
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e ized Brink-Axel (gBA) hypothesis for γ transitions between states in the quasi-continuum. The
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gBAhypothesisholdsnotonlyforspecificcollectiveresonances,butforthefulldipolestrength
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u below the neutron separation energy. We discuss the validity of the gBA hypothesis also for
n lightersystemslike92Zrwheretheconceptofauniqueγ-raystrengthfunction(γSF)isproblem-
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aticduetolargePorter-Thomasfluctuations. MethodsforstudyingtheγSFandthefluctuations

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asfunctionofexcitationenergyarepresented.
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The26thInternationalNuclearPhysicsConference
11-16September,2016
Adelaide,Australia
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
TheBrink-Axelhypothesis M.Guttormsen
1. Introduction
More than sixty years ago, Brink proposed in his doctoral thesis at Oxford University [1, 2]
thatthephotoabsorptioncrosssectionofthegiantelectricdipoleresonance(GDR)isindependent
of the detailed structure of the initial state. The hypothesis was further extended by the principle
ofdetailedbalance,toincludeabsorptionandemissionofγ raysbetweenresonantstates[3,4]. In
moregeneralterms,thisgeneralizedBrink-Axel(gBA)hypothesisimpliesthatthedipoleγ-decay
strengthhasnoexplicitdependenceonexcitationenergy,spinorparity,excepttheobviousdipole
transitionselectionrules.
ThegBAhypothesisisfrequentlyappliedinavarietyofapplicationsasitdramaticallysimpli-
fies the considered problem and in some cases, is a necessity in order to perform the calculations.
Hence, the question of whether the hypothesis is valid or not, and under which circumstances, is
of utmost importance. For this discussion, one should keep in mind that the original formulation
ofthehypothesiswasmeantformoderateenergies,ascitedfromBrink’sthesis: "Nowweassume
thattheenergydependenceofthephotoeffectisindependentofthedetailedstructureoftheinitial
statesothat,ifitwerepossibletoperformthephotoeffectonanexcitedstate,thecrosssectionfor
absorption of a photon of energy E would still have an energy dependence given by (15)", where
the equation (15) refers to a Lorentzian shape of the photoabsorption cross section on the ground
state.
The gBA hypothesis has fundamental impact on nuclear structure and dynamics, and has a
pivotal role in the description of γ and β decay for applied nuclear physics. In particular, the
hypothesis is often used for calculating (n,γ) cross-section needed to model the astrophysical r-
processnucleosynthesisandthenext-generationoffastnuclearreactors.
Experimental and theoretical information on the validity of the gBA hypothesis is rather
scarce. Primary transitions from (n,γ) reactions give γSFs consistent with the gBA hypothesis,
but only in a limited γ-energy range [5, 6, 7, 8, 9]. On the other hand, the 89Y(p,γ)90Zr reac-
tion points towards deviations from the gBA hypothesis [10]. There have also been theoretical
attempts to test the gBA hypothesis. Here, deviations and even violations to the hypothesis are
found[11,12]. However,forothertheoreticalapplications,theassumptionofthegBAhypothesis
issuccessfullyapplied[13,14,15,16,17].
Fromtheabovementionedfindings,itisnotatallobviouswhenthegBAhypothesisisvalid.
This unclear situation is not only due to peculiar structures and dynamics of the systems studied,
but also in some cases, due to huge Porter-Thomas fluctuations [18] that may camouflage the un-
derlyingphysics. Inthiswork,wewillshowthatatmoderateexcitationenergiesandwithaproper
averaging over many γ transitions, the gBA hypothesis is a fruitful concept for the exploration of
thenuclearquasi-continuumregion.
2. Methodandtools
TheOslomethodhasproventobearobusttechniquetoextractgrosspropertiesintheenergy
region below the particle separation energy in terms of nuclear level density (NLD) and γ-ray
strength function (γSF) [19, 20]. The method is based on measuring the γ decay after light ion
reactionswhereonlyonechargedparticleisejected. Inthisway,onemaycollectγ spectratagged
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TheBrink-Axelhypothesis M.Guttormsen
5 5
4 4 110022 110022
V)
Me 3 237Np(d,pg )238Np 3
(Ei
2 2 1100 1100
a) unfolded b) primary
1 1
0 0 11 11
1 2 3 4 5 1 2 3 4 5
E (MeV) E (MeV)
g g
Figure 1: (Color online) Initial excitation energy E versus γ-ray energy E from particle-γ coincidences
i γ
recorded with the 237Np(d,pγ)238Np reaction [23, 24]. The γ-ray spectra unfolded by the NaI response
function(a)andtheprimaryorfirst-generationγ-rayspectraP(E ,E)(b)areextractedasfunctionofinitial
γ i
excitationenergyE.
i
withtheexcitationenergy. Suchcoincidencedataarethenorganizedintoan(E ,E )matrix,which
γ x
represents the basis for the Oslo method. Without going into details, this matrix is then corrected
for the detector response function [21] and then utilized to extract the first-generation (primary) γ
spectra[22]. Forillustration,theunfoldedandprimaryγ matricesof238NpareshowninFig.1
The primary γ-ray matrix, which represents the energy distribution of the first emitted γ-ray
transitionineachcascade(seeFig.1(b)),isassumedtobefactorizedby
P(E ,E)∝ρ(E −E )T (E ). (2.1)
γ i i γ γ
The proportionality to ρ(E −E ) is in accordance with the Fermi’s golden rule [25, 26], and the
i γ
second factor, the γ-ray transmission coefficient T , is assumed to be independent of excitation
energyaccordingtotheBrinkhypothesis[1]. Fordipoletransitions,T relatestotheγSFby
1 T (E )
γ
f(E )= . (2.2)
γ 2π E3
γ
In the standard Oslo method, we utilize the γ spectra above a certain initial excitation region
(in the 238Np case, E =3.03 MeV) where the decay is assumed to be of statistical nature. The
min
upperexcitationregionislimitedtotheneutronseparationenergybyE ≈S .
max n
Byacceptingtheleveldensityobtainedwiththefitofρ andT toalargepartofthePmatrix
(the standard Oslo method), we may investigate the transmission coefficient in more detail, and
thusthevalidityoftheBrinkhypothesis. WeadoptthesolutionsT andρ fromEq.(2.1)andwrite
P(E ,E)
T (E ,E)≈N (E) γ i . (2.3)
γ i i
ρ(E −E )
i γ
withanormalizationfunctiongivenby
(cid:82)EidE T (E )ρ(E −E )
N (E)= 0 γ γ i γ . (2.4)
i (cid:82)EidE P(E ,E)
0 γ γ i
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TheBrink-Axelhypothesis M.Guttormsen
Six decay possibilities
for dipole transitions
5-
4+ 5+ 5- 6-
4- 6+
Figure2: (Coloronline)Theleveldensityextractedfromthe92Zr(p,p(cid:48))92Zrreactionbasedontheprimary
P(E ,E) matrix. In general, the dipole transition in the quasi-continuum goes with six transitions: two
γ i
stretchedM1,twostretchedE1,oneun-stretchedM1andoneun-stretchedE1transition.
SinceN onlydependsonE,wemaynowstudyhowT (andthus f)variesasfunctionofE for
i γ
eachexcitationbin. AnexpressionforT (E ,E )canbeconstructedinthesamewayasabove[24].
γ f
ThefluctuationsexpectedfortheγSFareassumedtofollowtheχ2 distributionwhereν isthe
ν
numberoftransitionsnincludedintheaveragingforacertainenergyE . Therelativefluctuations
γ
(cid:112)
of the χ2 distribution are given by r = 2/ν. With the experimental information on the level
ν
density ρ (see e.g. Fig. 2), we may count the number of transitions expected from an initial to a
final excitation energy bin. For the standard Oslo method, a large part of the primary γ matrix
is utilized to obtain a good averaging. Here, all the rows from E to E of the P matrix are
min max
included:
Emax 1
n(E )=∆E2 ∑ ∑ ∑ ∑ρ(E,I,π)·ρ(E −E ,I+L,π(cid:48)), (2.5)
γ i i γ
Ei=Emin Iπ L=−1 π(cid:48)
where ∆E is the energy-bin width for initial and final excitation energies. If we assume the decay
fromonlyoneinitialexcitationenergybin,thenumberoftransitionsisgivenby
1
n(E ,E)=∆E2∑ ∑ ∑ρ(E,I,π)·ρ(E −E ,I+L,π(cid:48)). (2.6)
γ i i i γ
Iπ L=−1 π(cid:48)
A similar expression can be constructed for the cases when the decay ends at a certain final ex-
citation energy E =E −E . The two formulae above have to be taken with care if not all spins
f i γ
andparitiesareavailable. Inparticular,forthedirectdipoledecaytothe0+ groundstate,onlyone
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TheBrink-Axelhypothesis M.Guttormsen
(a) E = 3.52 MeV 238Np (a) E = 6.22 MeV 92Zr
i i S
10−7 n
10−6 Data Data
E
Standard Standard
i
10−7
10−8
10−8
1 2 3 4 5 1 2 3 4 5 6 7 8
(b) E = 4.49 MeV (b) E = 7.42 MeV
i i
10−7
-3 )V 10−6 -3 )V
e e
M M
F (10−7 F (
γ S γ S10−8
10−8
1 2 3 4 5 1 2 3 4 5 6 7 8
(c) E = 5.46 MeV (c) E = 8.62 MeV
i i
-3 )V -3 )V 10−6 10−7
e e
M M
F ( F ( 10−7
S S 10−8
γ γ
10−8
1 2 3 4 5 1 2 3 4 5 6 7 8
E (MeV) E (MeV)
γ γ
Figure 3: (Color online) Gamma-ray strength functions of 238Np (left) and 92Zr (right) extracted from
various initial excitation bins E (see illustration). The data points with error bars are from statistically
i
independent data sets. The blue lines are obtained by the standard Oslo method where a large part of the
primaryγ-raymatrixhasbeenexploitedgivingabetteraveragingoftheγSFbyincludingmoretransitions.
Weobservesignificantfluctuationsinthe92Zrcase.
transitionappears,eitheronestretchedM1orE1transition. AsillustratedinFig.2,thisiscontrary
tosixtransitionsinthehighleveldensityregion.
3. Discussion
In the following we will discuss the γ decay in the quasi-continuum of 238Np and 92Zr. Both
nucleihavebeenmeasuredwithhighstatisticsallowingustodrawconclusionsonthedependence
oftheγSFswithexcitationenergy. Theodd-odd238Npnucleushasanextremelyhighleveldensity
of ≈43 million levels per MeV at S =5.488 MeV. Also at low excitation energy of ≈200 keV
n
wefind≈200levelsperMeV.Thus,forthisultimatesystem,wedonotexpectsignificantPorter-
ThomasfluctuationsandthegBAhypothesiscanbetested. However,the92Zrnucleusrepresentsa
systemwith≈10000timeslesslevelsthan238Npmakingitmoredifficulttodrawconclusionson
thevalidityofthegBAhypothesis.
Figure 3 shows γSFs from three initial excitation bins by use of Eq. (2.3). The blue curve is
basedonthestandardOslomethodofEq.(2.1)includingmanymoreprimaryγ transitions. Inthe
caseof238Np,theindividualγSFsindeedfollowthesamefunctionobtainedwiththestandardOslo
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TheBrink-Axelhypothesis M.Guttormsen
E_i = 3.03 - 5.46 MeV 238Np E_i = 4.54 - 8.62 MeV 92Zr
10-1 E_i = 5.46 MeV 102 E_i = 8.62 MeV
%) %)
s ( s (
n n
atio 10-2 atio 10
u u
ct ct
u u
Fl Fl
10-3 1
10-04.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10-1 1 2 3 4 5 6 7 8
g energy E (MeV) g energy E (MeV)
g g
Figure 4: (Color online) Relative fluctuations r(E ) in the γ strength of 238Np (left) and 92Zr (right) as
γ
function of E . The red lines are evaluated for a narrow initial high-energy excitation bin, and the blue
γ
curvesaretheresultsbasedonabroadinitialexcitationregion(standardOslomethod).
method. The close resemblance with a constant excitation-independent γSF for these three initial
excitationsisfoundforallthe22excitationbinsbetween3.03and5.46MeV(notshown).
As expected, the 92Zr nucleus reveals strong fluctuations where the individual γSFs deviate
more from the standard Oslo method predictions (blue curve) than given by the statistical uncer-
tainties. Since this nucleus has lower NLD, we even find for certain E -values that none or very
γ
fewγ transitionsexistwithinthegiveninitialexcitationgateE. Asanexample,intherightlower
i
panel(c)ofFigure3adropisseenintheγSFsforE ≈8.0MeVsincenolevelexistsin92Zrata
γ
finalexcitationenergyofE =E −E =8.62−8.00MeV=0.62MeV.
f i γ
Thefindingsof92Zrversus238NpraiseaneedforamorequantitativepredictionofthePorter-
Thomas fluctuations. If there is an insufficient averaging of the number of γ transitions, the fluc-
tuations could be large and the individual γSFs should then be different. However, this may not
invalidatethegBAhypothesis,whichpredictsanunderlyingdirectivefortheγ-decayprobabillity.
InordertoestimatethePorter-Thomasfluctuations, weapplytheexpressionsforthenumber
oftransitionsdescribedforthestandardOslomethodinEq.(2.5)andforaspecificinitialexcitation
energyinEq.(2.6). Therelativefluctuationsr(E )ofFig.4areseentoincreaseexponentiallywith
γ
E for both nuclei. We also note that the relative fluctuations are 2 - 3 times less for the standard
γ
Oslo method compared to the fluctuations using an individual narrow initial excitation bin. The
fluctuations in 238Np are extremely low going from r ≈0.0001% at low E and up to ≈0.1% at
γ
S . The 92Zr nucleus shows higher fluctuations than the statistical errors, and up to r ≈40 % at
n
E ≈8 MeV. The individual γSFs of 92Zr (only 3 out of 34 shown in Fig. 3) fluctuate around the
γ
γSFsobtainedwiththestandardOslomethod,andthussupportthevalidityofthegBAhypothesis
alsoforthislightnucleus.
4. Conclusion
We have studied the γSF between excitation energy bins in 238Np and 92Zr. The γ decay in
238Npisindependentontheexcitationenergy,andthusvalidatesthegBAhypothesis. Inthiscase,
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TheBrink-Axelhypothesis M.Guttormsen
thehypothesisworksnotonlyforspecificcollectiveresonances,butforthewholedipoleγSFbelow
theneutronseparationenergy.
AnecessityfortestingthevalidityofthegBAhypothesisisthatthePorter-Thomasfluctuations
are smaller than the experimental errors. This means that the sampling of the γSF should include
sufficiently many γ transitions. A technique has been developed using the measured level density
toestimatethenumberoftransitionswithacertainγ energy.
For the case of 92Zr, large fluctuations are found that apparently contradict the occurence of
oneuniqueγSFforallexcitationenergies. Acloserexaminationrevealsthattheselargedeviations
appearduetofewornon-existinglevelsforagivenE atcertainexcitationbins. Inparticular,huge
γ
fluctuationsappearforγSFsincludinglow-lyingstates,e.g. theIπ =0+ groundstate. Theindivid-
ualγSFsseemtofluctuatearoundacommonγSF,whichgivessupporttothegBAhypothesisalso
forlighternuclei.
Acknowledgements
We would like to thank E. A. Olsen, J. C. Müller, A. Semchenkov, and J. C. Wikne for providing
high quality experimental conditions. A. C. L. gratefully acknowledges funding through ERC-
STG-2014 under grant agreement no. 637686. G. M. T. gratefully acknowledges funding of this
researchfromtheResearchCouncilofNorway,ProjectGrantNo. 222287.
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