Low-energy limit of the radiative dipole strength in nuclei
Elena LitvinovaDepartment of Physics, Western Michigan University, Kalamazoo, Michigan 49008-5252, USA and National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, Michigan 48824-1321, USA
Nikolay BelovNuclear Physics Department, St. Petersburg State University, 198504 St. Petersburg, Russia
(Received 28 February 2013; revised manuscript received 13 August 2013; published 4 September 2013)
We explain the low-energy anomaly reported in several experimental studies of the radiative dipole strengthfunctions in medium-mass nuclei. These strength functions at very low γ energies correspond to the γ transitionsbetween very close nuclear excited states in the quasicontinuum and attract an increasing interest because of theirsubstantial astrophysical impact. We show that the low-energy enhancement of the strength functions in highlyexcited compound nuclei is explained by nucleonic transitions from the thermally unblocked single-quasiparticlestates to the single-(quasi)particle continuum. The case of radiative dipole strength functions at the nuclearexcitation energies typical for the thermal neutron capture is illustrated for 94,96,98Mo and 116,122Sn in comparisonto available data.
Experimental and theoretical studies of the nuclear low-energy electric dipole response remain among the challengesof the modern nuclear structure physics and attract an in-creasing interest because of its astrophysical impact. Radiativestrength (γ strength) at low energies may enhance the neutroncapture rates in the r process of nucleosynthesis [1,2] with aconsiderable influence on elemental abundance distributions.One of the key phases of the r-process nucleosynthesis iscapture of a thermal neutron with the subsequent γ decayof the compound nucleus. The typical neutron energy inthe astrophysical plasma is about 100 keV. Therefore, thedescription of γ -emission spectra of a compound nucleuswith excitation energies of the order of the neutron separationenergy is the central problem. The Hauser-Feshbach model isa standard tool for calculations of the radiative neutron capturecross sections [3]. Formally, this model includes all possibledecay channels via transmission coefficients. In the γ -decaychannel the corresponding coefficient is determined by theradiative strength function, which is usually calculated by oneof the phenomenological parametrizations [4–6]. However, inmore recent works [1,2,7] it has been shown that for the mostimportant electric dipole strength these simple models are notsufficient because they do not account for structural details ofthe strength at the neutron threshold. Sensitivity of the stellarreaction rates to these details emphasizes the importance oftheir studies within microscopic self-consistent models.
Another key ingredient for the Hauser-Feshbach calcu-lations is the Brink-Axel hypothesis [8] stating that the γstrength does not depend on the nuclear excitation energy,in particular, it is the same for excited and nonexcited nuclei.Supposedly true for the giant resonances and for the soft modessuch as pygmy dipole resonance, this hypothesis is, however,violated for the lowest transition energies. For instance,nonzero strength is systematically observed at very low γenergies [9]. Radiative strength functions (RSFs) extractedfrom various measurements [10–14] show an upbend at γenergies Eγ � 3 MeV in light nuclei of Fe-Mo mass region.
Studies of Ref. [15] have revealed that this phenomenon,occurring in various astrophysical sites, leads to one to twoorders of magnitude enhancement of the radiative neutroncapture rates for exotic neutron-rich nuclei, which, in turn,results in up to 43% average change of the final r-processabundances [16]. Phenomenological approaches approximatethe low-energy γ strength by the tail of the giant dipole reso-nance with a temperature-dependent width. This is, however,not justified, because the low-energy γ strength originatesfrom underlying physics, which is completely different fromthe giant vibrational motion. Modern microscopic theorieshave excellent tools for computing probabilities of transitionsbetween the nuclear ground state and excited states, but havecommon problems to describe transitions between excitedstates. The general many-body techniques such as Green’sfunction formalism [17,18], can be applied to γ emissionand γ absorption in excited states of compound type ifit is approximated by a semistatistical model, such as afinite-temperature mean field. In such a case, the transitionsare described by the finite-temperature version of the randomphase approximation (RPA) and its extensions. There existformulations within discrete model spaces [19–21] and modelswith exact treatment of the single-particle continuum [22–25].
In particular, a comprehensive study of γ emission fromhot rotating nuclei has been performed in Refs. [26,27]where behavior of the low-frequency γ strength in nucleiof lanthanide region is discussed in the context of collectiveversus statistical decays. The intrinsic response function iscalculated within the Hartree-Fock-Bogoliubov model andRPA in the discrete model space and transformed to thelaboratory frame. The resulting E1, M1, and E2 strengthfunctions show upbends at Eγ → 0 already at very moderatetemperatures.
In this work, we further explore the mechanism forthe enhancement of the low-frequency dipole γ transitionsbetween the nuclear excited states in the quasicontinuum.We show that in spherical nuclei this phenomenon can be
ELENA LITVINOVA AND NIKOLAY BELOV PHYSICAL REVIEW C 88, 031302(R) (2013)
quantitatively described in terms of a microscopic many-body approach built on the thermal mean-field descriptionof the compound nucleus. Exact treatment of single-particlecontinuum at finite temperature and exact elimination of thecenter-of-mass motion are the two essential ingredients forunderstanding these dipole γ transitions with frequenciesEγ � 3–4 MeV.
The general concept of the finite-temperature mean-fieldtheory [19,20,28] is based on the variational principle ofmaximum entropy minimizing the thermodynamical potential
�(λ, T ) = E − λN − T S, (1)
with the Lagrange multipliers λ and T determined by theaverage energy E, particle number N , and the entropy S. Thesequantities are the thermal averages involving the generalizedone-body density operator R:
S = −kTr(RlnR), E = Tr(RH), N = Tr(RN ), (2)
where H is the nuclear Hamiltonian, N is the particle numberoperator, and k is the Boltzmann constant. Varying the Eq. (1),one can determine the density operator R with the unity trace:
R = e−(H−λN )/kT
Tr[e−(H−λN )/kT ], H = δE[R]
δR . (3)
For definiteness, we start from a spherical even-evencompound nucleus with spin and parity 0+. We describe γemission and γ absorption as an interaction of the nucleus witha sufficiently weak external electromagnetic field P oscillatingwith some frequency ω. The interaction with such a fieldcauses small amplitude nuclear oscillations around the staticequilibrium, so that the total density matrixRhas an oscillatingterm, in addition to the static thermal mean-field part R0:
R(t) = R0 + [δRe−iωt + H.c.]. (4)
Variation δR of the density matrix R in the external fieldP obeys, in the local approximation, the following integralequation:
δR(x; ω, T ) = δR(0)(x; ω, T ) +∫
dx ′dx ′′A(x, x ′; ω, T )
×F (x ′, x ′′)δR(x ′′; ω, T ), (5)
where x is a multi-index x = {r, s, τ, χ} of spatial coordinater, spin s, isospin τ , and component in the quasiparticlespace χ . F (x, x ′) is the effective nucleon-nucleon interaction,A(x, x ′; ω, T ) is the two-quasiparticle propagator in thenuclear medium at finite temperature and
δR(0)(x; ω, T ) =∫
dx ′A(x, x ′; ω, T )P (x ′). (6)
The propagator A(x, x ′; ω, T ) is the key quantity and ideallyhas to include all the in-medium and surface effects. In thefirst approximation we calculate it within the thermal contin-uum quasiparticle random phase approximation (TCQRPA)in terms of the Matsubara temperature Green’s functions[17,18]. The full expression of the TCQRPA propagator inthe coordinate space is presented in Ref. [24], for the caseof spherical symmetry. The propagator consists of the discreteand continuum parts. The discrete part describes transitions be-tween the single-quasiparticle states in the discrete spectrum,
Continuum
E
n(E)
E
n(E)
T > 0 T = 0
εF
FIG. 1. (Color online) Schematic picture of the possible lowest-energy single-quasiparticle transitions from the thermally unblockedstates in an excited compound (left) and from the frozen ones in theground-state nucleus (right).
and the continuum part describes transitions from the discretespectrum states to the continuum. Dashed and solid arrowsin Fig. 1 show the low-frequency transitions of both kinds,respectively. For the case of γ emission, the arrows indicatephotons while nucleons transit back to lower-energy orbits.For the absorption the situation is reversed. The effectiveoccupation probability distribution ni(Ei, T ) has much largerdiffuseness at finite T than at T = 0, being the followingproduct: ni(Ei, T ) = v2
i (T )[1 − ni(Ei, T )] below the Fermienergy εF , and ni(Ei, T ) = [1 − v2
i (T )]ni(Ei, T ) above εF ,vi are the occupation numbers of Bogoliubov quasiparticles,
ni(Ei, T ) = 1
1 + exp[Ei(T )/kT ], (7)
and Ei are the eigenvalues of the single-particle Hamiltonian.The mean field is generated by the Woods-Saxon (WS)potential providing realistic description of the single-particlestates and the effective nucleon-nucleon interaction F (x, x ′)has the Landau-Migdal ansatz. The dipole radiative strengthfunction (RSF) corresponding to the 0+ → 1− transition isdetermined by the quantity δR through its convolution withthe electromagnetic dipole operator PE1(x) = eτ rY1(r) witheffective charges en = −Z/A, ep = N/A:
fE1(Eγ , T ) = − 8e2
27(hc)3Im
∫dxP
†E1(x)δR(x; ω, T ), (8)
ω = Eγ + i, → 0. Formally, Eq. (8) corresponds to γabsorption, and γ -emission strength can be calculated for thefinal temperature Tf = √
(E∗ − δ − Eγ )/a [29]. However, forEγ � 3–4 MeV the γ absorption and γ -emission strengthfunctions are close to each other. Their differences will bediscussed elsewhere.
The spurious translational mode is eliminated exactly aswe discuss the strength shape below 3–4 MeV where usuallythis mode appears because of numerical inaccuracies even inself-consistent versions of QRPA. Therefore, it is unavoid-able to apply a forced consistency procedure changing theHamiltonian accordingly [30]. This means that self-consistent
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FIG. 2. (Color online) Radiative dipole strength in 122Sn calcu-lated within TCQRPA with different smearing parameters, see textfor details.
models lose their advantage in the present case and explainsour choice of the approach.
Figure 2 shows the radiative dipole strength in 122Snat finite and zero temperature computed with diminishingsmearing parameters , so that the absence of admixture ofthe Goldstone mode is clearly demonstrated. Moreover, whileat T = 0 the low-energy strength is just an artificial tail of thefirst excited state of the discrete spectrum, at finite temperaturethe low-energy strength has the pure continuum origin andpractically saturated at = 10 keV. At this and smaller valuesof the finite-temperature strength at low energies showssteps at the energies equal to the energies of the single-particlestates closest to the continuum εi = εF + Ei , which confirmsthe interpretation given by Fig. 1.
For a more detailed illustration of finite temperature effects,we have selected some nuclei for which the dipole RSFhave been studied recently and reported in Refs. [11,31,32].Figures 3 and 4 display the dipole RSF in 94,96,98Mo and116,122Sn calculated within the TCQRPA at finite and zerotemperatures, compared to data. To be specific, in this work weconsider the nuclear excitation energy E∗ equal to the neutronseparation energy E∗ = Sn. The corresponding temperatureparameter T is determined from the phenomenological relationT = √
(E∗ − δ)/a, where δ is the so-called back shift and ais the level density parameter. For both δ and a there areno universal values. The numerical values for δ are takenfrom Ref. [33]. For a we have taken the values from theenhanced generalized superfluid model [33] as upper limitsand the lower limits are obtained microscopically from thesingle-particle level densities of neutrons gν and protons gπ
in the WS potential as a = π2(gν + gπ )/6. Thus, the intervalsof relevant temperatures vary from nucleus to nucleus as 1.26� T � 1.59; 1.15 � T � 1.55; 1.02 � T � 1.52 MeV for94,96,98Mo and 1.03 � T � 1.3, 1.02 � T � 1.17 MeV for116,122Sn, respectively. The uncertainty in determining thetemperature parameter corresponds to the uncertainty in thedata normalization discussed in Ref. [15] for Mo isotopes.The colored bands in Fig. 3 bordered by the strengths at min-
FIG. 3. (Color online) The E1 γ -strength functions for even-evenMo isotopes at finite temperatures obtained within the TCQRPA,compared to data [11,15] and to the γ strength for the ground state(T = 0, dash-dotted curves).
imal and maximal temperature parameters can be comparedwith data obtained with both normalization procedures andagree reasonably within the bands, however, the uncertaintiesfor the temperature are too large to be in favor of a particularnormalization. While the dash-dotted blue curves (T = 0)
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FIG. 4. (Color online) Same as in Fig. 3, but for 116,122Sn,compared to data from Refs. [31,32].
show at low γ energies the tails of the higher-energy transitionsonly due to the nonzero value of , at finite T there isthe pure thermal continuum strength, which remains finite at → 0, as explained in Fig. 2. This means that the low-energystrength has the origin, which is completely different from thehigh-frequency nuclear oscillations. The results obtained forodd-even Mo isotopes are similar to that for the even-evenones and on the same level of agreement to data.
RSF in tin isotopes are chosen as a counterexample andshown in Fig. 4. The calculated RSF in 116,122Sn show noupbends at the relevant temperatures (pink bands in Fig. 4).The reason is that the upper temperature limits given aboveare smaller than in Mo isotopes due to the larger WS valuesof a.
Figure 1 gives a qualitative interpretation for the low-energyenhancement of the γ strength. It is clearly seen that transitionsfrom the thermally unblocked states of the single-particlespectrum to the continuum (solid arrow in Fig. 1, left partT > 0) form solely the γ -strength function at very lowtransition frequency Eγ . Such types of transitions are notpossible in the ground state (T = 0) where the lowest-energytransitions to the continuum have much higher energies (solid
arrow in Fig. 1, right part). This schematic picture also explainswhy the low-energy RSF grows with temperature, however, thenumerical calculations show that the precise behavior of thestrength depends on the particular details of the single-particlestructure.
Although thermal QRPA with exact continuum treatmentmight explain the origin and the leading mechanism of for-mation of the γ strength at low energies, one should be awarethat this approach is considered rather as the starting point on away to a more complete theory of nuclear RSF. The mechanismdiscussed in the present Rapid Communication alone producesthe γ -strength functions with very smooth variations with themass number stemming from the level density although largervariations with the particle number are seen in the experimentalRSF. The general experimental observation is that lighternuclei such as Mo, Fe, Sc, V show the upbend while onesheavier than Sn do not [15]. A rather smooth evolution of theRSF both within particular isotopic chains [32] and withinvarious nuclei in a certain mass region [15] is observed, takinginto account the uncertainties of the measurements and dataextraction. Nevertheless, deviations from the regular behaviorreflect structure peculiarities of the individual nuclei caused byincoherent collisionlike interactions. Although quantitativelythese effects are expected to be of the next order at very lowenergies, they have to be taken into consideration.
Mechanisms like coupling to complex configurations[34,35] and thermal fluctuations [36] cause resonancelikestructures on the strength functions at energies above 4–5 MeVand should be included on top of the TCQRPA. Furthermore,the grand canonical description implies a statistical descriptionmixing several nuclei (with different number of protons andneutrons), which is not bad for modest temperatures but whichmakes the approach unreliable at the high-temperature limits ofthe present calculations (strictly speaking, the upper limits ofthe considered temperatures calculated from the microscopiclevel density are too high and, in fact, such high temperaturesmay compensate effectively the deficiency of the TCQRPAconfiguration space). The theoretical strength functions ofFigs. 3 and 4 are very smooth while the experimental onespresent larger variations with the particle number. Therefore,the theory should be considerably changed to perform acanonical description and to go beyond the QRPA to includefurther correlations and to improve the level density since theQRPA one is not the right one at these excitation energies.Ideally, the theory has to include all correlations, continuum,and finite temperature. However, the finite-temperature mi-croscopic model including continuum beyond QRPA doesnot exist. Existing theories beyond QRPA either miss exactcontinuum or developed only for zero temperature. In thisRapid Communication our goal is, however, not to build acomplete theory, but to explain the physical mechanism of theobserved upbend. We have shown that the transitions betweenthermally unblocked discrete states and continuum can be theleading mechanism for the formation of the continuous γstrength below 3–4 MeV of γ energy, but a more completemodel is needed for nuclear RSF.
Summarizing, we give a theoretical interpretation of thelow-energy anomaly in the behavior of the radiative dipolestrength in medium-mass and heavy nuclei. We have shown
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that a microscopic approach to nuclear response with cou-pling to the continuum and exactly eliminated center-of-mass motion, based on the statistical description of thecompound nucleus, gives a very good first approximationto the low-energy γ strength already on the level of thetwo-quasiparticle configurations. Application to electric dipoleresponse explains the systematic low-energy enhancementof the γ strength on the microscopic level. Thus, it isshown that microscopic nuclear many-body theory can bebrought to the domain which was previously dominatedby phenomenological approaches. The obtained results have
an important consequence for astrophysics, namely for theapproaches to r-process nucleosynthesis: those involvingBrink-Axel hypothesis may need to be revised.
The authors are indebted to H. Feldmeier, Yu. Ivanov, E.Kolomeitsev, G. Martınez-Pinedo, V. Tselyaev, and V. Zelevin-sky for enlightening discussions. Support from HelmholtzAlliance EMMI and from NSCL (E.L.), from GSI SummerStudent Program 2009, and from the St. Petersburg State Uni-versity under Grant No. 11.38.648.2013 (N.B.) is gratefullyacknowledged.
[1] S. Goriely and E. Khan, Nucl. Phys. A 706, 217 (2002).[2] S. Goriely, E. Khan, and M. Samyn, Nucl. Phys. A 739, 331
(2004).[3] J. J. Cowan, F.-K. Thielemann, and J. W. Truran, Phys. Rep.
208, 267 (1991).[4] S. G. Kadmenskii, V. P. Markushev, V. I. Furman, Yad. Fiz. 37,
277 (1983) [Sov. J. Nucl. Phys. 37, 165 (1983)].[5] J. Kopecky and M. Uhl, Phys. Rev. C 41, 1941 (1990).[6] S. F. Mughabghab and C. L. Dunford, Phys. Lett. B 487, 155
(2000).[7] E. Litvinova et al., Nucl. Phys. A 823, 26 (2009).[8] D. Brink, Ph.D. thesis, Oxford University, 1955 (unpublished);
P. Axel, Phys. Rev. 126, 671 (1962).[9] A. Schiller and M. Thoennessen, At. Data Nucl. Data Tables 93,
549 (2007).[10] A. Voinov et al., Phys. Rev. Lett. 93, 142504 (2004).[11] M. Guttormsen et al., Phys. Rev. C 71, 044307 (2005).[12] A. C. Larsen et al., Phys. Rev. C 76, 044303 (2007).[13] E. Algin et al., Phys. Rev. C 78, 054321 (2008).[14] M. Wiedeking et al., Phys. Rev. Lett. 108, 162503 (2012).[15] A. C. Larsen and S. Goriely, Phys. Rev. C 82, 014318 (2010).[16] R. Surman, J. Beun, G. C. McLaughlin, and W. R. Hix, Phys.
Rev. C 79, 045809 (2009).[17] T. Matsubara, Prog. Theor. Phys. 14, 351 (1955).[18] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods
of Quantum Field Theory in Statistical Physics (Prentice-Hall,Upper Saddle River, New Jersey, 1963).
[19] P. Ring et al., Nucl. Phys. A 419, 261 (1983).[20] H. M. Sommermann, Ann. Phys. (NY) 151, 163
(1983).[21] Y. F. Niu et al., Phys. Lett. B 681, 315 (2009).[22] J. Bar-Touv, Phys. Rev. C 32, 1369 (1985).[23] V. A. Rodin and M. G. Urin, Phys. Part. Nuclei 31, 975
(2000).[24] E. V. Litvinova, S. P. Kamerdzhiev, and V. I. Tselyaev, Yad. Fiz.
66, 584 (2003) [Phys. At. Nucl. 66, 558 (2003)].[25] E. Khan, N. Van Giai, and M. Grasso, Nucl. Phys. A 731, 311
(2004).[26] J. L. Egido and H. A. Weidenmuller, Phys. Lett. B 208, 58
(1988).[27] J. L. Egido and H. A. Weidenmuller, Phys. Rev. C 39, 2398
(1989).[28] A. L. Goodman, Nucl. Phys. A 352, 30 (1981).[29] V. A. Plujko, Nucl. Phys. A 649, 209c (1999).[30] S. Kamerdzhiev, R. J. Liotta, E. Litvinova, and V. Tselyaev,
Phys. Rev. C 58, 172 (1998).[31] H. K. Toft et al., Phys. Rev. C 81, 064311 (2010).[32] H. K. Toft et al., Phys. Rev. C 83, 044320 (2011).[33] http://www-nds.iaea.org/RIPL-3/[34] E. Litvinova, P. Ring, and V. Tselyaev, Phys. Rev. C 78, 014312
(2008).[35] V. Tselyaev et al., Phys. Rev. C 75, 014315
(2007).[36] M. Gallardo et al., Nucl. Phys. A 443, 415 (1985).
