Determining (n,f) cross sections for actinide nuclei indirectly

20 Pages·2006·1.37 MB·English

UCRL-JRNL-221555
Determining (n,f) cross sections for
actinide nuclei indirectly: An examination
of the Surrogate Ratio Method
J. E. Escher, F. S. Dietrich
May 23, 2006
Physical Review C
Disclaimer

This document was prepared as an account of work sponsored by an agency of the United States
Government. Neither the United States Government nor the University of California nor any of their
employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for
the accuracy, completeness, or usefulness of any information, apparatus, product, or process
disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any
specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise,
does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United
States Government or the University of California. The views and opinions of authors expressed herein
do not necessarily state or reflect those of the United States Government or the University of California,
and shall not be used for advertising or product endorsement purposes.
Determining (n,f) cross sections for actinide nuclei indirectly:
An examination of the Surrogate Ratio Method
Jutta E. Escher∗ and Frank S. Dietrich†
Lawrence Livermore National Laboratory, Livermore, CA 94550
(Dated: May 22, 2006)
ThevalidityoftheSurrogateRatiomethodfordetermining(n,f)crosssectionsforactinidenuclei
is examined. This method relates the ratio of two compound-nucleus reaction cross sections to a
ratio of coincidence eventsfrom two measurements in which thesame compound nuclei are formed
via a direct reaction. With certain assumptions, the method allows one of the cross sections to
be inferred if the other is known. We develop a nuclear reaction-model simulation to investigate
whether theassumptions underlying theRatio approach are valid and employ these simulations to
assess whether the cross sections obtained indirectly by applying a Ratio analysis agree with the
expectedresults. Inparticular,wesimulateSurrogateexperimentsthatallowustodeterminefission
cross sections for selected actinide nuclei. The nuclei studied, 233U and 235U, are very similar to
thoseconsideredinrecentSurrogateexperiments. WefindthatinfavorablecasestheRatiomethod
providesusefulestimatesofthedesiredcrosssections, andwediscusssomeofthelimitationsofthe
approach.
PACSnumbers: 25.85.Ec,24.60.Dr,24.50.+g,27.90.+b
I. INTRODUCTION It is useful to consider whether certain simplifications
or approximations to the method can be utilized to
determine relevant cross sections. A simple, approxi-
Nuclearreactiondataplayanimportantroleinnuclear
mate version of the Surrogate technique, which we will
physics applications. Unfortunately, for a large number
refer to as the “Surrogate Method in the Weisskopf-
of reactions the relevant data cannot be directly mea-
Ewing limit” was already used in the 1970s to estimate
sured in the laboratory or reliably predicted by calcula-
neutron-inducedfissioncrosssectionsfromtransferreac-
tions. Direct measurements of reactions on unstable iso-
tions[8–12]andhasreceivedrenewedattentioninrecent
topes are particularly affected since the relevant nuclei
years [13]. In the work presented here, we will focus on
are often too difficult to produce with currently avail-
a related approximation, the so-called “Surrogate Ratio
able experimental techniques or too short-lived to serve
Method” or simply the “Ratio Method”.
astargetsinpresent-dayset-ups. Calculationsarehighly
The purpose of this paper is an examination of the
nontrivial since they typically require a thorough under-
validity of the Surrogate Ratio method for determining
standing of both direct and statistical reaction mecha-
(n,f) cross sections for actinide nuclei. The study was
nisms (as well as their interplay) and a detailed knowl-
motivatedbyrecent(d,d0f)and(α,α0f)Surrogateexper-
edgeofthenuclearstructureinvolved. Itisthereforeim-
imentsatYale[14]andBerkeley[15],respectively,which
portanttoexplorealternativeapproachesfordetermining
were analyzed in the framework of the Ratio approach.
reaction cross sections on unstable nuclei.
Inthepresentstudywewilluseanuclearreaction-model
The “Surrogate Reaction Method” is a specific indi-
simulation to investigate whether the assumptions un-
rectmethodthatcombinesexperimentwithreactionthe-
derlying the Ratio method are valid, and employ these
ory to obtain cross sections for compound-nucleus reac-
simulations to assess whether the cross section obtained
tions involving difficult-to-produce targets. While the
indirectly by applying the Ratio method agrees with the
Surrogate method is very general and can in principle
expected result. We will also comment on the validity
be employed to determine cross sections for all types of
of the Surrogate Method in the Weisskopf-Ewing limit,
compound-nucleus reactions involving a large variety of
since it is closely related to the Ratio approach.
target nuclei, there are various challenges that have to
The experiments described in [14, 15] investigated in-
be addressedinorderto validateandimplement this ap-
directly neutron-inducedfissionfor 235U and237U, while
proach for the different regions of interest in the nuclear
we chose to focus on (n,f) reactions on 233U and 235U.
chart. Forapplicationsto(n,f)crosssectionsonactinide
Thischoicehastheadvantagethatwestudynucleiwhich
nuclei,YounesandBritthavestudiedsomeoftheseissues
are very similar to those consideredin the recent experi-
recently[1,2]. Forapplicationsto(n,γ)crosssectionson
ments,butforwhichallofthe relevantcrosssectionsare
lighter nuclei, in particular on s-process branch points,
known from direct measurements. Therefore, the results
work is currently underway [3–7].
of the present study can be used to reach conclusions
aboutthe accuracyofthe Ratio technique formeasuring
the previously unknown 237U(n,f) cross section.
∗Electronicaddress: [email protected] This document is organized as follows: In the next
†Electronicaddress: [email protected] section, we will introduce the main concepts employed
2
in the present report. In particular, we will explain the Weisskopf-EwinglimitofthefullHauser-Feshbachtheory
Surrogatemethod, review the Weisskopf-Ewingapproxi- is briefly reviewed in the context of Surrogate reactions
mationtothemethod,andintroducetheRatioapproach. andtheRatioMethod,whichmakesuseoftheSurrogate
In Section III we describe the simulations we are using idea and assumes the validity of the Weisskopf-Ewing
and outline the logic of the tests we are carrying out in approximation, is introduced.
order to assess the validity of the Ratio method. In Sec-
tion IV we presentcalculations that test the primaryas-
sumption underlying the Ratio approach, the validity of A. The Surrogate Idea
the Weisskopf-Ewing approximation. We then compare
cross section predictions of both the Surrogate Method TheSurrogatenuclearreactiontechniqueisanindirect
in the Weisskopf-Ewing limit and the Surrogate Ratio methodfordeterminingthe crosssectionfora particular
method to predetermined reference cross sections. Our type of “desired” reaction, namely a two-step reaction,
findings and conclusions are summarized in Section V. a +A → B∗ → c+ C, that proceeds through a com-
pound nuclear state B∗, a highly excited state in statis-
tical equilibrium (see Figure 1).
II. SURROGATE APPROACHES The formalism appropriate for describing compound-
nucleus reactions is the statistical Hauser-Feshbach the-
This section introduces the main concepts employed ory (see, e.g., chapter 10 of Ref. [16]). The averagecross
in this study. The Surrogate idea is explained and the section per unit energy in the outgoing channel χ0 for
challenges associated with carrying out, analyzing, and reactions proceeding to an energy region in the final nu-
interpreting a Surrogate experiment are outlined. The cleus described by a level density is given by:
dσαHdχEF(χE0 α) =πλ2αXJπ ωαJlsXl0s0I0 P0χ00l00s00TχJ00l00s00(ETχα0J0l)s+(EPα)χT0χ0Jl00l00ss000I(E00Rχ0T)ρχJI000l(0U0s000)(Eχ00)ρI00(U00)dEχ00. (1)
Here α denotes the entrance channel a+A with energy integralof transmissioncoefficients andleveldensities in
E and reduced wavelength λ . The spin of the inci- the residual nuclei). The relevant quantitites in these fi-
α α
dent particle is i, the target spin is I, the channel spin nal channels χ00 are denoted by double-primed symbols,
is ~s = ~ı + I~, and the compound-nucleus angular mo- in analogy to the particular channel of interest, χ0. In
mentum and parity are J,π. The excitation energy of writing Eq. 1, we have suppressed the parity quantum
the compound nucleus, E , is related to E via the number except for that of the compound nucleus. The
ex α
separation energy S (B) of the particle a in the nu- leveldensitydepends inprincipleonparity(eventhough
a
cleus B: E = S (B) + E . The transmission coef- this dependence is weak in practice), and all sums over
ex a α
ficient associated with the entrance channel is denoted quantum numbers must respect angular-momentumand
TJ and the statistical-weight factor ωJ is given by parity conservation.
αls α
(2J +1)/[(2i+1)(2I+1)]. Quantitites associated with The above Hauser-Feshbach formula neglects correla-
the exit channel of interest are denoted by primed sym- tions between the incident and outgoing reaction chan-
bols: χ0 = c0+C0, i0 is the spin of the outgoing particle nels that can be taken into account formally by includ-
c0, I0 is the spin of the residual nucleus C0,~s0 =~ı0+I~0 is ing width fluctuation corrections. These correlationsen-
the channel spin and Eχ0 the energy for χ0. The energy hance the elastic scattering cross section and reduce the
of the decaying nucleus, Eex, is related to Eχ0 via the inelastic and reaction cross sections, although this de-
relationEex =Sc0(B)+Eχ0, where Sc0(B) is the separa- pletion rarely exceeds 10-20% (even at energies below
tion energy of c0 in B. The transmission coefficients for approximately2MeV)andbecomesnegligible asthe ex-
this channel are written as TχJ0l0s0(Eχ0) and ρI0(U0) de- citationenergyofthecompoundnucleusincreases. Inthe
notes the density of levels of spin I0 at excitation energy remainderofthis study wewillneglectwidthfluctuation
U0 in the residual nucleus C0. All energetically possi- corrections and rewrite the Hauser-Feshbachformula as:
ble final channels χ00 haveto be taken into account,thus
dσHF(E )
the denominator includes contributions from decays to αdχE α = XσαCN(Eex,J,π)GCχN(Eex,J,π),(2)
discrete levels in the residual nuclei (given by the sum χ
Jπ
0) as well as contributions from decays to regions of
P
high level density in the residual nuclei (given by the where σαCN(Eex,J,π) denotes the cross section for form-
secondsuminthedenominatorwhichinvolvesanenergy ing the compound nucleus at excitation energyEex with
angular-momentum and parity quantum numbers Jπ in
3
the reaction a+A → B∗. The symbol GCN(E ,J,π) in coincidence with the outgoing particle b. The direct-
χ ex
is the branching ratio for the decay of this compound reactionparticle is typically stopped in a detector which
state into the desired exit channel χ. Here and in the providesparticleidentification, aswellasinformationon
remainderofthe paper wesuppressthe prime associated the kinetic energy and direction of b. The desired exit
with the exitchannelofinterest,unless itis necessaryto channelχcanbeidentified,e.g.,bydetectingfissionfrag-
distinguish explicitly between all possible final channels mentsfromB∗orγraysfromthedesiredresidualnucleus
(χ00) and the particular channel of interest (χ0). C. TheprobabilityforformingB∗ intheSurrogatereac-
In the limit of negligible width fluctuation corrections tion(withspecificvaluesforE ,J,π)isFCN(E ,J,π),
ex δ ex
considered here, the formation and decay of the com- whereδ referstothereactiond+D→b+B∗. Thequan-
pound nucleus are independent of each other, individ- tity
ually for each angular momentum and parity value. It
is this independence that allows one to determine the Pδχ(Eex)=XFδCN(Eex,J,π) GCχN(Eex,J,π), (3)
desiredcrosssectionvia acombinationoftheoryandex- J,π
periment in the Surrogate approach. In many cases the
which gives the probability that the compound nu-
formation cross section σCN can be calculated to a rea-
α cleus B∗ was formed with energy E and de-
sonable accuracy by using optical potentials while the ex
cayed into channel χ, can be obtained experimentally.
theoretical branching ratios GCN for the different chan-
χ The direct-reaction probabilities FCN(E ,J,π), where
nels χ are often quite uncertain. The objective of the δ ex
FCN(E ,J,π)=1,havetobedeterminedtheoret-
Surrogate method is to determine or constrain these de- PJπ δ ex
ically. In practice, the decay of the compound nucleus is
cay probabilities experimentally.
modeledand the GCN(E ,J,π)are obtainedby adjust-
χ ex
ing parameters in the model to reproduce the measured
decay probabilities P (E ). Subsequently, the branch-
δχ ex
ingratiosobtainedinthismannerareinsertedinEq.2to
(cid:9) (cid:10) (cid:11) (cid:12) (cid:13)(cid:14) (cid:11) (cid:15) (cid:16) (cid:14) (cid:11) (cid:17) (cid:18) (cid:19) (cid:13)(cid:20) (cid:21) yieldthe desiredreactioncrosssection. Inthe abovedis-
cussion,wehaveomittedtheangulardependenceofboth
(cid:0) (cid:1)
the desired and the Surrogate reactions on the observa-
(cid:2) (cid:3) tion angle of the particles emitted from the compound
nucleus B∗. The extension of the Hauser-Feshbach for-
mulae is straightforward[16].
(cid:5) In practice the procedure of determining the branch-
ing ratios is a difficult task due to several theoretical
(cid:4) and experimental challenges: i) The decay probability
P (E ) = N (E )/N (E ) is experimentally deter-
δχ ex δχ ex δ ex
mined: Here N denotes the total number of d+D →
δ
b + B∗ reactions, usually determined by observing the
(cid:9) (cid:22) (cid:23) (cid:14) (cid:14) (cid:20) (cid:24) (cid:17) (cid:19) (cid:11) (cid:16) (cid:14) (cid:11) (cid:17) (cid:18) (cid:19) (cid:13)(cid:20) (cid:21) outgoing particle b, and N is the number of b-χ co-
δχ
(cid:6) (cid:8) incidences. Both N and N need to be accurately
δχ δ
determined. If target contaminants are present, it be-
(cid:7)
comes very difficult, if not impossible to determine a
reliable value for N . ii) The theoretical prediction of
(cid:2) (cid:3) δ
the direct-reaction probabilities FCN(E ,J,π) requires
δ ex
a framework for calculating cross sections of direct re-
(cid:5) actions (stripping, pick-up, and inelastic scattering) to
continuum states in B∗. iii) Extracting the branching
ratiosGCN(E ,J,π) frommeasureddecay probabilities
(cid:4) χ ex
P (E ) requires modeling the decay of the compound
δχ ex
nucleus produced in the Surrogate reaction and fitting
FIG. 1: Schematic representation of the “desired” (top) and the relevant parameters to reproduce the experimental
“Surrogate” (bottom) reaction mechanisms. The basic idea
results. iv)The possibility thatthe intermediatenucleus
oftheSurrogateapproachistoreplacethefirststepofthede- B∗ producedintheSurrogatereactiondecaysbeforesta-
siredreaction,a+A,byanalternative(“Surrogate”)reaction,
tisticalequilibriumisreachedhastobe excludedormin-
d+D→b+B∗,thatpopulatesthesamecompoundnucleus.
imized.
The subsequent decay of the compound nucleus into the rel-
evant channel can then be measured and used to extract the
desired cross section.
B. The Weisskopf-Ewing Limit
In a Surrogate experiment, the compound nucleus B∗
is produced via an alternative (“Surrogate”), direct re- The Hauser-Feshbach theory rigorously conserves to-
action d+D →b+B∗ and the decay of B∗ is observed tal angular momentum J and parity π. Under certain
4
conditions the branching ratios GCN(E ,J,π) can be cross section. Calculating the direct-reaction probabili-
χ ex
treated as independent of J and π and the form of the ties FCN(E ,J,π) and modeling the decay of the com-
δ ex
cross section (for the desired reaction) simplifies to pound nucleus are no longer required. However, Surro-
gate experiments in the Weisskopf-Ewing limit are still
dσWE(E ) challenging since the requirement that both the number
αχ a = σCN(E )GCN(E ) (4)
dE α ex χ ex of b−χ coincidences and the number of reaction events
χ
be accurately determined remains. Although the cross
where section expressed in Eq. 4 is differential in the outgoing-
channel energy, the quantity of interest is the cross sec-
σαCN(Eex)=XσαCN(Eex,J,π) tion integrated over all final-state energies. In the fol-
J,π lowing, we will assume the quantity GχCN has been inte-
∞ gratedoverthe energyEχ ofthe final-statechannel,and
= πλ2αX(2l+1)Tαl(Eα) (5) will therefore remove the differentiation with respect to
l=0 energy in Eq. 4. The essential feature of this equation
remains;namely, the factorizationinto a formationcross
is the reaction cross section describing the formation of section and a branching ratio, neither of which depend
the compound nucleus at energy E and GCN(E ) de-
ex χ ex on angular momentum or parity.
notes the Jπ-independent branching ratio for the exit
channel χ. This is the Weisskopf-Ewing limit of the
Hauser-Feshbach theory. It is applicable when the fol-
C. The Ratio Approach
lowing conditions are satisfied [17, 18]:
• Theenergyofthecompoundnucleushastobesuffi- The “Ratio method” makes use of the Surrogate idea
ciently high,so thatalmostallchannelsinto which and requires the validity of the Weisskopf-Ewing limit.
the nucleus can decay are dominated by integrals An important motivation for using the Ratio method is
over the level density. the fact that it eliminates the need to accurately mea-
sure N , the total number of d+D → b+B∗ reaction
δ
• Width fluctuations have to be negligible. This will events, which has been the source of the largest uncer-
be the case if the previous condition is satisfied. tainty inSurrogateexperiments performedrecently. Un-
der the proper circumstances it also reduces or removes
• The transmission coefficients TχJ00l00j00 associated dependence on the angular distribution of fission frag-
with the available exit channels have to be inde- ments, which is not well characterizedin the present ex-
pendent of the spin of the states reached in these periments.
channels. Thisconditionissufficientlywellsatisfied ThegoaloftheRatiomethodistodeterminetheratio
sincethedependenceoftransmissioncoefficientson
target spin is very weak. σ (E)
R(E) = α1χ1 (6)
σ (E)
• The level densities ρI00(U00) in the available chan- α2χ2
nels haveto be independent of parity andtheir de-
of the cross sections of two compound-nucleus reactions,
pendence on the spin I00 of the relevant nuclei has a +A →B∗ →c +C and a +A →B∗ →c +C ,
to be of the form ρI00(U00) ∝ (2I00+1). It can be w1here t1he tw1o reac1tions1have to2 be “2simila2r” in a2 sens2e
shown that for sufficiently high energies U00, level
thatremainstobespecified. Anindependentdetermina-
densities are very weakly dependent on parity, so
tionofoneofthesecrosssectionsthenallowsonetoinfer
thatthefirstoftheseconditionscanbe assumedto
the other by using the ratio R. In the Weisskopf-Ewing
be satisfied. The second condition, which is a pre-
limit, the ratio R(E) can be written as:
requisiteforarigorousderivationoftheWeisskopf-
Ewing limit from the full Hauser-Feshbach theory, σCN(E)GCN(E)
is satisfied if the spin I00 is smaller than the spin R(E) = α1 χ1 , (7)
σCN(E)GCN(E)
cutoff parameter σ in the relevant level-density α2 χ2
cut
formula. In the actinide region σ ≈6−7, but it
cut with branching ratios GCN that are independent of J
is known that the Weisskopf-Ewing limit is still a χ
and π and compound-nucleus formation cross sections
useful approximationat higher spins.
σCN andσCN thatcanbecalculatedbyusinganoptical
α1 α2
The Weiskopf-Ewing limit providesa simple and pow- model.
erful approximate way of calculating cross sections for To determine GχC1N/GχC2N, two experiments are car-
compound-nucleus reactions. In the context of Surro- ried out. Both use the same direct-reaction mechanism,
gate reactions,it greatlysimplifies the applicationofthe D(d,b)B∗, but different targets, D1 and D2 to create
method: It becomes straightforward to obtain the Jπ- the relevant compound nuclei, B1∗ and B2∗, respectively.
independent branching ratios GCN(E ) from measure- For each experiment, the number of coincidence events,
χ ex
ments of P (E ) and to calculate the desired reaction N(1) andN(2) ,ismeasured. Theratioofthebranching
δχ ex δ1χ1 δ2χ2
5
ratios into the desired channels for the compound nuclei III. METHOD OF THE STUDY
created in the two reactions is given by
The purpose of the present work is an examination of
GCN(E) N(1) (E) N(2)(E) the validity ofthe SurrogateRatio methodfordetermin-
χ1 = δ1χ1 × δ2 . (8)
GχC2N(E) Nδ(22χ)2(E) Nδ(11)(E) iwnegw(nil,lf)i)curosessasHecatuiosnesr-fFoershabctaicnhid-beanseudclneiu.cIlenaprarretaicctuiloanr-,
modelsimulationtoinvestigatewhethertheprincipalas-
The experimental conditions are adjusted such that the
relative number of reaction events, norm = N(1)/N(2), sumptions underlying the Ratio method are valid, and
δ1 δ2 ii) employ these calculations to assess whether applying
can be determined by accountingfor differences in beam
a Ratio analysis to the simulated observables yields a
intensities and beam times, as well as numbers of atoms
crosssectionthatagreeswiththe expectedresult. While
ineachtarget. Thisrequiresthatthesamesetupbeused
experimentaltestscanbeveryvaluable,employingasim-
inbothexperiments. Theratioofthedecayprobabilities
ulationprovidesseveraldistinctadvantages: Weareable
thensimplyequalstheratioofthecoincidenceeventsand
toaccessphysicalquantitiesthatarenotdirectlyobserv-
R(E) becomes:
able in an experiment, such as Jπ-dependent branching
ratios for a specific exit channel. We can alter quan-
σCN(E)N(1) (E)
R(E) = α1 δ1χ1 , (9) tities that are experimentally not easily modified in a
σCN(E)N(2) (E) controlled manner, such as the angular-momentum dis-
α2 δ2χ2 tribution in a compound nucleus before it decays, and
where we havesetthe experimentalnormalizationfactor carry out sensitivity studies.
norm to unity. Forourstudy,weneedtwotypesofreferencecrosssec-
The definition of the energy E in the above equations tions: The first type, σα1χ1, appears in the denominator
remains to be specified. Typically, the energy depen- ofthe ratioR= σα1χ1 andcorrespondsto aknowncross
σα2χ2
dence of a compound-nucleus formation cross section, section in real-world applications of the Ratio method.
σCN = σ(a+A → B∗) is characterized by the kinetic The second type of reference cross section corresponds
α
energy of the projectile, E , while a branching ratio is to the unknown, “desired” cross section and serves as a
α
normally given as a function of the excitation energy benchmark for the Ratio method. We employ the full
of the compound nucleus, GCN(E ). In a compound- Hauser-Feshbach theory to calculate the reference cross
χ ex
nucleusreaction,thosetwovaluesarerelatedviathesep- sectionsrelevantforour study: 233U(n,f), 235U(n,f), and
arationenergyS oftheparticleainB∗: E =S +E . 235mU(n,f). We treat the calculated 233U(n,f) cross sec-
a ex a α
While either E or E can be used to uniquely specify tion as the reference cross section which will appear in
ex α
the energy dependence of such a reaction, it is impor- the denominator of the ratio R, while the 235U(n,f) and
tant for the Ratio method to make the comparison of 235mU(n,f) cross sections are to be determined from a
the relevant reactions, a + A → B∗ → c + C and Ratio treatment. Upon extracting these by applying the
1 1 1 1 1
a +A → B∗ → c +C at an energy that minimizes Ratioprescription,wecancomparetheresultstotheref-
2 2 2 2 2
uncertainties. Here, we take E to denote the kinetic en- erencecrosssectionsofthesecondtypeandgaininsights
ergy of the projectile. into the potential success of the Ratio method.
InRefs.[14]and[15],theRatiomethodwasusedtoob- Furthermore, to simulate a Surrogate treatment, we
tain an estimate of the 237U(n,f) cross section. Inelastic needto generatethose quantities that aretypically mea-
deuteron [14] and α [15] scattering experiments on 238U sured in a Surrogate experiment. In particular, we need
and236Uwerecarriedoutandfissionfragmentsfromthe observablesassociatedwiththedecayofacompoundnu-
decay of 238U∗ and 236U∗ were detected in coincidence cleus thatwasproducedina direct(Surrogate)reaction.
with the outgoing direct-reaction particle. The results Surrogate experiments focusing on (n,f) cross sections
were found to be in good agreement with a theoretical typically measure fission probabilities, while determing
estimate by Younes et al. [19]. (n,γ) cross sections involves measuring gamma-ray in-
More recently, a variant of the Ratio approach de- tensitiesfortransitionswithintheresidualnucleus. Both
scribed here was explored for determining the 237U(n,γ) fission probabilities and gamma intensities can be calcu-
and 237U(n,2n) cross sections [20]. Rather than compar- lated with a Hauser-Feshbach code, such as the Stapre
ingthedecayoftwocompoundnuclei,B∗andB∗,formed code [21, 22] used in the present work. In order to ac-
1 2
withthesametypeofSurrogatereaction,theauthorsde- countfor the fact that a direct reactionproduces a spin-
terminedtheratioofdecayprobabilitiesfortwodifferent parity distribution in the compound nucleus that is dif-
exit channels, χ = c +C and χ = c +C , of one ferent from the spin-parity distribution associated with
1 1 1 2 2 2
particular compound nucleus, B∗ =B∗ ≡B∗. This is an the desired reaction, we modified this code so that it is
1 2
interestingapproachthatdeservesfurtherstudy,whichis suitable for Surrogate-reactionstudies. In particular, we
outside the scope of the present paper. We will restrict includedanoptiontoallowthespin-paritydistributionof
our considerations to applications of the Ratio method the first compound nucleus to be read in from a text file
in which two similar compound nuclei are formed in two rather than calculated from the entrance-channel trans-
separate Surrogate reactions. mission coefficients. It is therefore possible to select an
6
arbitrary Jπ distribution for a given compound nucleus
and to predict the decay of the nucleus.
In Section IV, we will discuss various tests of the
Ratio method for fission. We begin by investigating
the main assumption of the method, the validity of
the Weisskopf-Ewing limit, by explicitly considering the
branching ratios GCN(E ,J,π) of Equation 2, where χ
χ ex
refers to the fission channel. In particular, we study the
GCN(E ,J,π) values that result from the fitting proce-
χ ex
dureofSubsectionIIIAbelow,fordifferentspinandpar-
ity values Jπ as a function of energy. We then simulate
aSurrogatefissionexperiment. Specifically,westudyfis-
sionof234U and236U followingexcitationbya directre-
action. Rather thanattempting to predictthe Jπ distri-
butions that can result from the various possible direct-
reaction mechanisms (transfer and inelastic reactions),
we consider a few simple spin-parity distributions for
234Uand236Uandobservetheireffectonthecompound-
nucleus decay. We carry out a Surrogate analysis under
the assumption that the Weisskopf-Ewing limit is valid
FIG. 2: Compound-nuclear cross section obtained from the
and infer the σ(235U(n,f)) and σ(235mU(n,f)) cross sec- Flap2.2optical-modelpotential[23]. Thisisthereactioncross
tions. The results are compared to the reference cross section after removal of contributions from the excitation of
sections obtained from the full Hauser-Feshbach calcu- directly-coupledstates. Thecalculations werecarried outfor
lations. Furthermore, we use the fission simulation to neutronsincidenton234UandweassumeσnC+N234U =σnC+N235U
carryoutaRatioanalysisandcomparethecrosssections = σnC+N235mU = σnC+N233U.
obtained in this manner to the reference cross sections.
This comparison gives insight into the quality that can
beachievedwiththeRatiomethodgivenoptimalcircum- The compound-nucleus formation cross section σCN
n+235U
stances. = σCN = σCN is plotted in Figure 2. It is ex-
n+235mU n+233U
pected to be accurate to approximately 5% throughout
the energy range covered.
A. Nuclear-reaction model for fission
Both the calculations of the reference cross sections 1. Determining the reference cross sections
and the simulations of the Surrogate experiments make
use of the Hauser-Feshbach statistical reaction theory. We began this study with the results of an earlier
The former involve a standardHauser-Feshbachdescrip- treatment of the 235U(n,2n) that had been carried out
tion of the formation and decay of a compound nucleus, with the stapre code [24]. In this earlier calculation,
with parameters adjusted to reproduce available exper- levelschemes, level densities, gamma strength functions,
imental data. The simulations of the Surrogate exper- fission-modelparameters,andpreequilibriumparameters
iments employ the Hauser-Feshbach theory to describe had been carefully adjusted to fit the available experi-
the decay of the relevant compound nucleus only, with mental data on gamma production and fission. For the
parameters taken from the reference-cross section calcu- presentwork,onlysmalladjustments weremaderelative
lations. The determination of the level schemes, level to the earlier calculation. The calculation of neutron-
densities, gamma strength functions, fission-model pa- induced fission of 233U carried out for the present study
rameters,andpreequilibriumparametersisdescribedbe- proceeded in a nearly identical manner. Since we are in-
low. terested in comparing reactions on the two targets, it is
Thefollowingtargetnucleiareconsideredinourstudy: important that the procedures applied to both be very
233U which has ground-state spin and parity Iπ = 5+, similar. The following remarks are applicable to both
2
235UwhichhasIπ = 7−,and235mUwithIπ = 1+. Since calculations.
2 2
weareinterestedinneutron-inducedreactions,appropri-
ate neutron-nucleus optical potentials are needed. For • Level schemes and gamma branching ratios were
simplicity, the calculations employ the same deformed taken from the evaluated nuclear-structure data
neutron-nucleus optical potential for all three systems, available on the RIPL-2 web site [25]. Level den-
i.e. the neutron transmission coefficients are indepen- sities were parameterized in the Gilbert-Cameron
dent of the target nucleus. This is a suitable approx- form [26]. The Fermi-gas parameter was fit to the
imation since the optical-model observables vary very observed average s-wave spacing, or obtained from
slowly over the range of uranium isotopes considered. the Gilbert-Cameron systematics if experimental
7
data were unavailable. The constant-temperature
portionwasconstrainedtoreproducethenumberof
observeddiscretelevelsatlowenergy,andmatched
in value and slope to the Fermi-gas form at an en-
ergy somewhat below the neutron separation en-
ergy.
• Detailsofthedeformedopticalpotential(‘Flap2.2’)
usedinthecurrentcalculationsaregivenin[23];its
predicted reaction cross section is shown in Fig. 2.
• Transmissioncoefficientsforfissionwerecalculated
from the usual two-barrier model (see, for exam-
ple, Bjornholm and Lynn [27]) in which the fission
process proceeds through transition states built
on top of the two saddle points. In the present
calculations the transition states were represented
by a level density; no discrete transition states
were considered. For use in the fission calcula-
tions, an additional option for calculating level-
densities was added to stapre. This option allows
four constant-temperature segments, connected to
a Fermi-gas form for use at energies above the
constant-temperature segments. The level densi-
ties are required to be continuous at the match-
ing points. Separate spin-cutoff parameters may
bespecifiedforthe constant-temperaturesegments
and for the Fermi-gas region. The spin-cutoff pa-
rameters used in the fits to the fission data were
inthe range5–7,consistentwiththose in Ref.[27].
The starting values for the fission-barrier heights
andcurvatureswerealsotakenfromBjornholmand
Lynn[27]. Theseaswellasthelevel-densityparam-
eters were varied to achieve fits to the fission cross
section data.
• The gamma transmission coefficients were calcu-
latedfromastandardBrink-Axelmodel[28,29]us-
ing a double-humped Lorentzian parameterization
ofthegiantdipoleresonance,togetherwithasmall FIG.3: Fitstothe233U(toppanel)and235U(bottompanel)
M1 component with a Weisskopf (E3) energy de- fission cross sections.
γ
pendence. Exactly the same parameters were used
in all nuclei considered; the strength (peak GDR
results shown in this report.
crosssection)oftheE1componentwasadjustedto
reproduce the average of the experimental values
The results of the fits to the fission cross sections of
for the radiationwidth of the s-waveresonances in
233U and 235U are displayed in Fig. 3. The calculated
the uranium isotopes.
first-, second-, and third-chance fission as well as their
• Preequilibrium neutron emission in the first (n,n0) sum are shown, together with the evaluations to which
the fission parameters were fitted. The ENDL-99 [30]
stage of the reaction was calculated with the ex-
citon model built into the stapre code. Its pa- evaluation and a beta version of the ENDF-7 [31] were
used for 235U and 233U, respectively.
rameters were determined in the earlier study of
reactions on 235U by requiring that the preequilib- Basedonthesuccessfulfitoftheexperimental(oreval-
uated)crosssectionsfor233Uand235U,thecrosssections
rium spectrum be in approximate agreement with
known (n,n0) data in the actinide region. Exactly for reactions on the first-excited (isomeric) state of 235U
can be predicted by changing the spin and parity of the
thesameparameterswereusedinallofthepresent
target from 7/2− to 1/2+ and keeping all other param-
calculations for both isotopes.
eters fixed. The prediction for 235mU(n,f) is shown in
• Width-fluctuation correctionswere not included in Fig.4,alongwiththeevaluationforfissionoftheground
any of these calculations, nor in any of the other state. At the lowest energies, the cross section is lower
8
compound-nuclearJπ distributions andalso providedif-
ferentchallengesforapropertheoreticaldescription. For
instance, for the reactions of interesthere, both inelastic
αanddeuteronscatteringfromactinidenuclei,aswellas
the (3He,α) transferreaction,havetobe considered,and
apropertreatmentofthe deformationisrequired. Thus,
ratherthanfocusingonobtainingareliablepredictionof
the Jπ distributions that can result for these cases, we
will use schematic distributions that are likely cover the
range of relevant Jπ distributions.
Some guidance for selecting possible Jπ distributions
can be found in the literature. One-nucleon strip-
ping reactions, such as (α,3He), (20Ne,19Ne), etc., with
intermediate-energyprojectiles,E ≈20-40MeV/A,have
receivedmuchinterestinrecentyears. Experimental[32–
34] as well as theoretical [35] work has been devoted to
understanding the continuum spectra from such single-
nucleon stripping reactions on spherical nuclei. In the
reactionsconsidered,relativelylargeangular-momentum
transfers(l =5,6,7)resulted inthe populationof high-j
FIG.4: Predictionof235mUfissioncrosssectionusingthepa- single-particle orbitals (1h ,1i , 1j , etc.), which
rameters obtained from thefit to the235U(n,f) cross section. were a major focus of thos1e1s/t2udi1e3s/.2Cal1c5u/2lations for the
For reference purposes, the black line shows the evaluated 239Pu(d,p)240PureactionatE =13MeV[36]andforthe
235U(n,f) cross section. d
two-nucleon stripping reaction (t,p) at E = 15-18 MeV
t
for the actinide region [1] have shown smaller angular-
momentum transfers to be relevant as well. Since reac-
thanthatonthegroundstatebyabout20%. Thisreduc- tions involving targets with large ground state spins can
tionisinqualitativeagreementwiththeresultsofYounes
produce compound-nuclear Jπ distributions peaked at
and Britt [1], even though the present calculation does largerangular-momentumvalues,itisreasonabletocon-
not include the discrete transition states on top of the
siderJ >10aswell. Furthermore,inelasticscatteringon
fissionbarriersthatarebelievedtoplayasignificantrole even-eventargetsexhibitsacharacteristicasymmetrybe-
in the difference between the ground and isomeric cross
tweennaturalandunnaturalparitystates: Intheabsence
sections. of a spin-dependent interaction only Jπ= 0+,1−,2+,...
are populated (in a DWBA description). It is therefore
worthwhile to study the effects of Jπ distributions that
2. Simulating Surrogate experiments contain natural (or unnatural) parity states only.
For the purposes of the present study, we consider six
different distributions. The first four, distributions a,
In orderto simulate the decayof the compoundnuclei
b, c, and d, assume equal probabilities for positive and
234U and 236U, as produced in a Surrogate experiment,
negative parity states, and are plotted in Figure 5 (top
we employ the Hauser-Feshbach parameters determined
panel). Thesedistributionshavemeanangularmomenta
in the fits to the 233U(n,f) and 235U(n,f) cross sections,
hJi of 7.03, 10.0, 12.97, and 3.30 for the curves labeled
above. We also need information on the spin-parity dis-
a, b, c, and d, respectively. The last two distributions, e
tributions of the decaying 234U and 236U systems. In
and f, with mean angular momentum hJi=3.30, assume
principle, one would like to describe the direct-reaction
that only natural parity states (0+,1−,2+, etc.) or only
process (transfer or inelastic scattering reaction) lead-
unnatural-parity states (0−,1+,2−, etc.) are occupied,
ingtothesecompoundnucleiwithanappropriatedirect-
respectively,see Figure 5 (bottom panel). In Section IV,
reaction model and obtain information on the resulting
the dependence of the calculated branching ratios and
Jπdistribution,whichinturncanbeusedasinputinthe
extracted cross sections on the Jπ distributions of the
modified Hauser-Feshbach code. However, this is a non-
relevantcompoundnucleuswillbeinvestigatedusingthe
trivial task since it requires a description of transfer and
schematic forms introduced here.
inelastic scattering reactions leading to unbound states,
as well as an understanding of preequilbrium decay fol-
lowingadirectreaction(seeSectionIIIB2below). More-
B. The role of preequilibrium decays
over, a variety of projectile-target combinations with a
rangeofpossible incidentenergiesmaybe consideredfor
the direct reaction that produces the compound nucleus Central to the Surrogate approach is the assumption
ofinterest. Differentreactionmechanisms,regionsofthe thattheformationanddecayoftheintermediatenuclear
nuclear chart, and projectile energies will yield different state – in both the “desired” and the Surrogate reaction