Nuclear Astrophysics requires the knowledge of reaction rates over a wide
range of nuclei and temperatures. In recent calculations the nuclear level
density - as an important ingredient to the statistical model (Hauser-Feshbach)
- has shown the highest uncertainties. In a back-shifted Fermi-gas formalism
utilizing an energy-dependent level density parameter and employing microscopic
corrections from a recent FRDM mass formula, we obtain a highly improved fit to
experimental level densities. The resulting level density is used for
determining criteria for the applicability of the statistical model on
neutron-induced reactions.Comment: 4 pages, uses espcrc1.sty, Proc. Intl. Conf. "Nuclei in the Cosmos
  IV", Univ. Notre Dame 1996, Nucl. Phys. A, in press. Postscript is also
  available at http://quasar.physik.unibas.ch/research.htm

Bethe

Cowan

Dean

F.-K. Thielemann

Fowler

Ignatyuk

Iljinov

K.-L. Kratz

Möller

Reisdorf

T. Rauscher

Wagoner

English

arXiv

Crossref

Nuclear Physics A

Applicability of the Hauser-Feshbach approach for the determination of astrophysical reaction rates

Rauscher, T.

Thielemann, F. K.

Kratz, K. L.

edoc

Nuclear Astrophysics requires the knowledge of reaction rates over a wide range of nuclei and temperatures. In recent calculations the nuclear level density – as an important ingredient to the statistical model (Hauser-Feshbach) – has shown the highest uncertainties. In a back-shifted fermi-Gas formalism utilizing an energy-dependent level density parameter and employing microscopic corrections from a recent FRDM mass formula, we obtain a highly improved fit to experimental level densities. The resulting level density is used for determining criteria for the applicability of the statistical modelPeer reviewe

Thielemann, Friedrich-Karl

Kratz, K.L.

University of Hertfordshire Research Archive

arXiv:astro-ph/9609012v1  2 Sep 19961Applicability of the Hauser-Feshbach approach for the determination ofastrophysical reaction ratesT. Rauschera∗, F.-K. Thielemanna, and K.-L. KratzbaInstitut fu¨r Physik, Universita¨t Basel,Klingelbergstr. 82, CH-4056 Basel, SwitzerlandbInstitut fu¨r Kernchemie, Universita¨t Mainz,Fritz-Strassmann-Weg 2, D-55099 Mainz, GermanyNuclear Astrophysics requires the knowledge of reaction rates over a wide range of nu-clei and temperatures. In recent calculations the nuclear level density – as an importantingredient to the statistical model (Hauser-Feshbach) – has shown the highest uncertain-ties. In a back-shifted fermi-Gas formalism utilizing an energy-dependent level densityparameter and employing microscopic corrections from a recent FRDM mass formula, weobtain a highly improved fit to experimental level densities. The resulting level density isused for determining criteria for the applicability of the statistical model.1. INTRODUCTIONThe field of Nuclear Astrophysics has to provide nuclear reaction rates suited for a widerange of astrophysical applications. Therefore, there is not only need for rates involvingall possible (stable and unstable) nuclei across the nuclear chart but also for temperaturesranging from 0 < T9 < 10 . For the majority of reactions, the statistical model (Hauser-Feshbach, SM) will be suited for the calculation and prediction of the rates. However,at low level densities the SM is not applicable anymore and single resonances and directcapture contributions have to be taken into account.For the user of such rates it would be convenient to have a simple means to determinethe validity of the SM approach, showing which rates need special attention and probablyfurther experimental investigation. In this work, we provide a “map” for the applicabilityof the SM depending on the interacting nuclei and the temperature.2. THE NUCLEAR LEVEL DENSITYConsiderable effort has been put into the improvement of the input for the SM calcula-tions (e.g. [1]). However, the nuclear level density still has shown the largest uncertaintiesamong the properties entering the SM. For calculating the level densities in the givencontext one does not only have to find reliable methods, but also computationally feasible∗APART fellow of the Austrian Academy of Sciences2ones. In dealing with thousands of nuclei one has to resort to simple models in order tominimize computer time.Such a simple model is the back-shifted Fermi-gas description [1,2], recently improvedby introducing an energy-dependent level density parameter a [3–5]. More sophisticatedMonte Carlo shell model calculations [6] have shown excellent agreement with this phe-nomenological approach and justified the application of the Fermi-gas description at andabove the neutron separation energy. Assuming equally distributed even and odd parities,one obtains the following form:ρ(U, J, pi) =12f(U, J)ρ(U) , (1)withρ(U) =1√2piσ√pi12a1/4exp(2√aU)U5/4, f(U, J) =2J + 12σ2exp(−J(J + 1)2σ2)(2)σ2 =Θrigidh¯2√Ua, Θrigid =25muAR2 , U = E − δ .An improved approach has to consider the energy dependence of the microscopic effectswhich are known to vanish at high excitation energies [5], i.e. the thermal damping ofmicroscopic effects. The level density parameter a is then described by [3]a(U,Z,N) = a˜(A)[1 + C(Z,N)f(U)U], (3)wherea˜(A) = αA+ βA2/3 (4)andf(U) = 1− exp(−γU) . (5)The shape of the function f(U) was found by approximation of numerical microscopic cal-culations based on the shell model. Thus, one is left with three open parameters, namelyα, β, and γ. The values of these parameters are determined by a fit to experimental s-waveneutron resonance spacing at the neutron separation energy [5]. The values α = 0.1336,β = −0.06712, γ = 0.04862 result in a highly improved fit with an averaged global devia-tion of 1.5 [7,8], when taking the microscopic corrections C(Z,N) from the latest FRDMmass formula [9] and consistently computing the backshift δ(Z,N)=1/2{∆n(Z,N) +∆p(Z,N)} with the neutron and proton pairing gaps ∆n,p from the same source.3. APPLICABILITY OF THE STATISTICAL MODELHaving found a suitable method to calculate level densities one can apply it to deter-mine the range of validity of the SM. It is often colloquially termed that the SM is onlyapplicable for intermediate and heavy nuclei. However, the only necessary condition forits application is a large number of resonances at the appropriate bombarding energies,so that the cross section can be described by an average over resonances.3 N Z Applicability of the Statistical Model (n)20 40 60 80 100 120 140 16020406080Figure1.Temperatures(inT9)forwhichthestatisticalmodelcanbeused.Plottedisthecompoundnucleusoftheneutron-inducedreaction.4The nuclear reaction rate per particle pair at a given stellar temperature T is deter-mined by folding the reaction cross section with the Maxwell-Boltzmann (MB) velocitydistribution of the projectiles [10]〈σv〉 =(8piµ)1/21(kT )3/2∫∞0σ(E)E exp(− EkT)dE . (6)An effective energy window is then found around the peak of the integrand at E0. Forcharged particles this is the so-called Gamow peak at E0 = E1/3G (kT/2)2/3 (with theGamow energy EG). For s-wave neutrons the effective peak coincides with the peak ofthe MB distribution at E0 = kT (close to the neutron separation energy), for higher partialwaves the energy window is shifted to slightly higher energies (similarily to the Gamowpeak) due to the centrifugal barrier [11]. The effective energy window for a given nucleusand temperature has to contain sufficiently many resonances in order to make it possibleto solve the integral with the assumption of an average level density instead of calculatingthe exact sum over the individual levels. Numerical test calculations [8] have shown thatan average number of 5–10 contributing resonances is sufficient. Choosing a lower limitfor the number of resonances, determining the width and location of the effective energywindow at a given temperature and using the above level density description to calculatethe number of resonances in this window, we derive a lower limit for the temperatureat which the SM can still be used. Those temperature limits are shown in Fig. 1 forneutron-induced reactions. It should be noted that the derived temperatures will notchange considerably even if changing the required level number within a factor of two,because of the exponential dependence of the level density on the excitation energy.4. SUMMARYMaking use of an improved level density description we presented a method to determinethe applicability of the SM, also providing clues on which reactions might be of specialinterest for experimental investigations. In principle, the method can be used for any pro-jectiles and also with different incident energy distributions (e.g. for experimental beams).REFERENCES1. J.J. Cowan, F.-K. Thielemann and J.W. Truran, Phys. Rep. 208 (1991) 267.2. A.H. Bethe, Phys. Rev. 50 (1936) 332.3. A.V. Ignatyuk, G.N. Smirenkin and A.S. Tishin, Yad. Phys. 21 (1975) 485.4. Reisdorf, Z. Phys. A 300 (1981) 227.5. A.S. Iljinov, M.V. Mebel et al., Nucl. Phys. A543 (1992) 517.6. D.J. Dean, S.E. Koonin, K. Langanke et al., Phys. Rev. Lett. 74 (1995) 2909.7. T. Rauscher, F.-K. Thielemann and K.-L. Kratz, Mem. Soc. Astr. Ital., in press.8. T. Rauscher, F.-K. Thielemann and K.-L. Kratz, to be submitted to Phys. Rev. C.9. P. Mo¨ller, J.R. Nix et al., At. Data Nucl. Data Tables 59 (1995) 185.10. W.A. Fowler, G.E. Caughlan and B.A. Zimmerman, Ann. Rev. Astron. Astrophys. 5(1967) 525.11. R.V. Wagoner, Ap. J. Suppl. 18 (1969) 247.

Applicability Of The Hauser-Feshbach Approach For The Determination of
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Applicability Of The Hauser-Feshbach Approach For The Determination of Astrophysical Reaction Rates

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