aa r X i v : . [ nu c l - t h ] M a r Transition sum rules in the shell model
Yi Lu
College of Physics and Engineering, Qufu Normal University,57 Jingxuan West Road, Qufu, Shandong 273165, ChinaSchool of Physics and Astronomy, Shanghai JiaoTong University, Shanghai 200240, China andDepartment of Physics, San Diego State University,5500 Campanile Drive, San Diego, CA 02182-1233, United States
Calvin W. Johnson
Department of Physics, San Diego State University,5500 Campanile Drive, San Diego, CA 02182-1233, United States (Dated: April 3, 2018)
Abstract
An important characterization of electromagnetic and weak transitions in atomic nuclei are sumrules. We focus on the non-energy-weighted sum rule (NEWSR), or total strength, and the energy-weighted sum rule (EWSR); the ratio of the EWSR to the NEWSR is the centroid or average energyof transition strengths from an nuclear initial state to all allowed final states. These sum rules can beexpressed as expectation values of operators, in the case of the EWSR a double commutator. Whilemost prior applications of the double-commutator have been to special cases, we derive generalformulas for matrix elements of both operators in a shell model framework (occupation space),given the input matrix elements for the nuclear Hamiltonian and for the transition operator. Withthese new formulas, we easily evaluate centroids of transition strength functions, with no need tocalculate daughter states. We apply this simple tool to a number of nuclides, and demonstratethe sum rules follow smooth secular behavior as a function of initial energy, as well as comparethe electric dipole (E1) sum rule against the famous Thomas-Reiche-Kuhn version. We also findsurprising systematic behaviors for ground state electric quadrupole (E2) centroids in the sd -shell. . INTRODUCTION Atomic nuclei are neither static nor exist in isolation. Their transitions play importantroles in fundamental, applied, and astro-physics, as well as revealing key information aboutnuclear structure beyond just excitation energies. In this paper we focus on electromagneticand weak transitions; such transition strength distributions are important for γ -spectroscopy,nucleosynthesis and ββ decays, as they are used to extract level densities [1], calculatenuclear reaction rates in stellar processes [2] and analyze ββ decay matrix elements [3].The strength function for a transition operator ˆ F from an initial state i at energy E i , toa final state f at absolute energy E f and excitation energy E x = E f − E i is defined as S ( E i , E x ) = X f δ ( E x + E i − E f ) (cid:12)(cid:12)(cid:12) h f (cid:12)(cid:12)(cid:12) ˆ F (cid:12)(cid:12)(cid:12) i i (cid:12)(cid:12)(cid:12) . (1)Sum rules are moments of the strength function, S k ( E i ) = Z ( E x ) k S ( E i , E x ) dE x . (2)Two of the most important sum rules, which we consider here, are S , the non-energy-weighted sum rule (NEWSR) or total strength, and S , the energy-weighted sum rule(EWSR). These sum rules provide compact information about strength functions. For ex-ample, the famous Ikeda sum rule [4] for Gamow-Teller (GT) transitions is the differencebetween the total β − strength and total β + strength: S ( GT − ) − S ( GT + ) = 3( N − Z ) g A , where g A is the axial vector coupling relative to the vector coupling g V . For investigationsof ‘quenching’ of g A [5], the NEWSR S can be a probe of the missing strengths due tohypothesized cross-shell configurations.The centroid of a strength distribution is just the ratio of the EWSR to the NEWSR, E centroid ( E i ) = S ( E i ) S ( E i ) . (3)For a compact distribution of a giant resonance, E centroid ( E i ) will be roughly the locationof the resonance peak, relative to the parent state energy E i ; of course, in the case ofhighly fragmented strength distributions this interpretation no longer holds, and in severelytruncated model spaces the centroid will be too low compared to experiment. Both theNEWSR S and E centroid ( E i ) can test the validity of the general Brink-Axel hypothesis[6, 7]. The general Brink-Axel hypothesis [8–10] assumes that the strength distribution oftransitions from any parent state is approximately the same, thus as a result E centroid ( E i )is independent on E i . Though it seems this hypothesis needs to be modified for E1[11–13],M1[14–16] (the low-energy γ anomaly) and GT[17] transitions, it is still being widely usedto calculate neutron-capture rates [18], extract nuclear level densities [1, 19, 20] and canhave a substantial impact on astrophysical relevance [2, 21].Sum rules are appealing not only because they characterize strength functions, but alsobecause using closure some sum rules can be rewritten as expectation values of operators[22]. Allowing for transition operators with good angular momentum rank K , one should2um over the z -component M , and the total strength S ( E i ) becomes X f X M |h f | ˆ F K,M | i i| = X M h i | ( ˆ F K,M ) † ˆ F K,M | i i . (4)Thus S ( E i ) can be easily evaluated numerically without calculating any final state. Thestrength sum can be used to evaluate the former mentioned Ikeda sum rule, useful as a checkon computations.The EWSR can be written as the expectation value of a double commutator, as long asthe transition operator behaves as a spherical operator under Hermitian conjugation [23], (cid:16) ˆ F KM (cid:17) † = ( − M ˆ F K, − M . (5)If we do not have (5), one cannot write the EWSR operator as a double commutator. Therequirement of this will have consequences when we look at charge-changing transition suchas β decay. In that case, one must include both β − and β + transitions.Invoking closure and Eq. (5), S ( E i ) becomes D i (cid:12)(cid:12)(cid:12) X M ( − M h ˆ F K, − M , [ ˆ H, ˆ F K,M ] i (cid:12)(cid:12)(cid:12) i E . (6)As an example, the Thomas-Reiche-Kuhn sum rule [24] evaluates the energy-weighted sumof E N electrons, and conserves to a constant proportionalto N/m e . In nuclear physics the corresponding sum rule is similar, though the EWSR isproportional to N Z/ Am N because the dipole is relative to the center of mass. Anotherexample is related with the “scissor mode” in rare-earth nuclei [25], for which the EWSRof low-lying ( < E X x B ( M
1; 0 +1 → + x ) E + x ∝ X x B ( E
2; 0 +1 → + x ) . (7)This EWSR is derived both in the IBM-2 model [26], and in the shell model [27, 28] withphenomenological interactions.One can compute sum rules with the Lanczos algorithm [29–32], which has a deep connec-tion to the classical moment problem. Given some initial state | Ψ i i , one applies an transitionoperator ˆ F and then uses ˆ F | Ψ i i as the pivot or starting state in the Lanczos algorithm. Thisrequires, however, one being able to carry out a matrix-vector multiplication in the Hilbertspace under consideration, which may not aways be possible or practical, for example in thecase of coupled clusters [33] or generator coordinate calculations [22, 34, 35]. Furthermore,for example in the M -scheme, or fixed J z , basis for the configuration-interaction shell model,if the initial state has angular momentum J i >
0, then applying an operator ˆ F K with an-gular momentum rank K will produce a state with mixed J f , with | J i − K | ≤ J f ≤ J i + K by the triangle rule. To compare to experiment, however, one generally needs a sum overfinal M values and average over initial M values, and to correctly use the Lanczos methodone must either do this explicitly or project out states of good angular momentum andextract strength functions via appropriate Clebsch-Gordan coefficients. This point is notemphasized in the literature. 3n this paper we go beyond specific cases and, in the next section, write down the generalform of the operators (4) and (6) in a spherical shell model basis. Although straightforward,the EWSR in particular is somewhat involved and to the best of our knowledge not published.Appendix A provide some of the details of derivation. In Ref. [36] we make available a C++code to generate those operator matrix elements. With such machinery one can directlycompute the NEWSR and EWSR easily for many nuclides and many transitions. Priorwork showed that the NEWSR follows simple secular behavior with the initial energy E i and gave a general argument [7]. In section III we show a few cases and also find simplesecular behavior. Finally, we illustrate the applicability by looking at systematics of groundstate E1 and E2 sum rules. II. FORMALISM AND FORMULAS
We work in the configuration-interaction shell model, using the occupation representation[37] with fermion single-particle creation and annihilation operators ˆ a † , ˆ a , respectively. Asis standard, our operators have good angular momentum. The labels of each single-particlestate include the magnitude of angular momentum j and z -component m ; there are otherimportant quantum numbers, in particular parity, orbital angular momentum l and label n for the radial wave function, but those values are absorbed into the values of matrixelements, so, for example, the details of our derivation are independent of whether or notone uses harmonic oscillator or Woods-Saxon or other single-particle radial wave functions.Because we are working in a shell model basis, we differentiate between single-particle states (labeled by j , m , and l, n, . . . ) and orbits , by which we mean the set of 2 j + 1 states withthe same j but different m . We assign fermion operators of different orbits different lower-case Latin letters: ˆ a † , ˆ b † , etc., to prevent a proliferation of subscripts. (In our derviations,when discussing generic operators, which may be single-fermion operators or composed ofproducts and sums of operators, we use lower-case Greek letters: α, β, . . . . ) In order to makeour results broadly usable, we will be slightly pedantic.To denote generic operators ˆ α, ˆ β coupled up to good total angular momentum J and total z -component M , we use the notation ( ˆ α ⊗ ˆ β ) JM . Hence we have the general pair creationoperator ˆ A † JM ( ab ) = (ˆ a † ⊗ ˆ b † ) JM , (8)with two particles in orbits a and b . We also introduce the adjoint of A † JM ( ab ), the pairannihilation operator, ˜ A JM ( cd ) = − (˜ c ⊗ ˜ d ) JM . (9)Here we use the standard convention ˜ c m c = ( − j c + m c ˆ c − m c , where m c is the z -componentof angular momentum; this guarantees that if ˆ a † jm transforms as a spherical tensor, so does˜ a jm [23]. An alternate notation isˆ A JM ( cd ) = (cid:16) ˆ A † JM ( cd ) (cid:17) † = ( − J + M ˜ A J, − M ( cd ) . (10)4ith this we can write down a standard form for any one- plus two-body Hamiltonianor Hamiltonian-like operator, which are angular momentum scalars. To simplify we useˆ H = X ab e ab ˆ n ab + 14 X abcd ζ ab ζ cd X J V J ( ab, cd ) X M ˆ A † JM ( ab ) ˆ A JM ( cd ) , (11)where ˆ n ab = P m ˆ a † m ˆ b m and ζ ab = √ δ ab . Here V J ( ab, cd ) = h ab ; J | ˆ V | cd ; J i is the matrixelement of the purely two-body part of ˆ H between normalized two-body states with goodangular momentum J ; because H is a scalar the value is independent of the z -component M . One can also write this, in slightly different formalism, as X ab e ab [ j a ] (cid:16) ˆ a † ⊗ ˜ b (cid:17) , + 14 X abcd ζ ab ζ cd X J V J ( ab, cd ) [ J ] (cid:16) ˆ A † J ( ab ) ⊗ ˜ A J ( cd ) (cid:17) , , (12)where we use the notatation [ x ] = √ x + 1, which some authors write as ˆ x (we use theformer to avoid getting confused with operators which always are denoted by either ˆ a or ˜ a ).Finally we also introduce one-body transition operators with good angular momentumrank K and z -component of angular momentum M ,ˆ F K,M = X ab F ab [ K ] − (cid:16) ˆ a † ⊗ ˜ b (cid:17) K,M . (13)Here F ab = h a || ˆ F K || b i is the reduced one-body matrix element using the Wigner-Eckarttheorem and the conventions of Edmonds [23]. For non-charge-changing transitions, Eq. (5)implies F ab = ( − j a − j b F ∗ ba . (14)With these definitions and conventions, we can now work out general formulas for sumrules. An important issue will be isospin. Realistic operators, such as M1, connect stateswith different isospin, and so rather than working in a formalism with good isospin we treatprotons and neutrons as being in separate orbits. (Counter to this, we give one examplewith isoscalar E2 transitions in section III.) A. Non-energy-weighted sum rules
The non-energy-weighted sum rule operator is given byˆ O NEW SR = ~F † · ~F = X M (cid:16) ˆ F KM (cid:17) † ˆ F KM = X M ( − M ˆ F K − M ˆ F KM , (15)using Eq. (5). Thenˆ O NEW SR = X ab ˆ n ab X c F ∗ ca F cb j a + 1 − X abcd F ∗ cb F ad X J (cid:26) j a j d Kj c j b J (cid:27) X µ ˆ A † Jµ ( ab ) ˆ A Jµ ( cd )= X ab (ˆ a † ⊗ ˜ b ) X c [ j a ] − F ∗ ca F cb − X abcd F ∗ cb F ad X J (cid:26) j a j d Kj c j b J (cid:27) [ J ] (cid:16) ˆ A † J ( ab ) ⊗ ˜ A J ( cd ) (cid:17) . (16)5y writing out the operator as an angular momentum scalar and to look “just like” aHamiltonian, for purposes of use in a shell-model code, we haveˆ O NEW SR = P ab g ab [ j a ]( a † ⊗ ˜ b ) , + P abcdJ ζ ab ζ cd W J ( ab, cd ) [ J ] (cid:16) A † J ( ab ) ⊗ ˜ A J ( cd ) (cid:17) , , (17)where the single-particle matrix element is g ab = X c F ∗ ca F cb j a + 1 . (18)We do not assume isospin symmetry, but assume our orbital labels also reference pro-tons/neutrons. So in (18) labels a and b must be the same, proton or neutron. Nowfor the two-body matrix elements: for identical particles in orbits (i.e., a, b, c, d all labelprotons or all label neutrons), we need to enforce antisymmetry, that is, W pp ( nn ) J ( ab, cd ) = − ( − j a + j b + J W pp ( nn ) J ( ba, cd ), etc: W pp ( nn ) J ( ab, cd ) = − P abJ ) ζ − ab ζ − cd (cid:26) j a j d Kj c j b J (cid:27) F pp ( nn ) ∗ cb F pp ( nn ) ad , (19)where P abJ = − ( − j a + j b + J P ab , and P ab is the exchange operator swapping a ↔ b . Herethe only terms in ˆ F which contribute are the non-charge-changing pieces, F pp and F nn .For proton-neutron interactions, where we assume labels a, c are proton and b, d areneutron, i.e., we want to compute W pnJ ( a π b ν , c π d ν ), we need to identify the proton-neutronparts of ˆ F . So we still have (18) and W pnJ ( ab, cd ) = − (cid:18)(cid:0) F pn ∗ cb F pnad + ( − j a + j b + j c + j d F np ∗ da F npbc (cid:1) (cid:26) j a j d Kj c j b J (cid:27) − ( − J (cid:0) ( − j a + j b F pp ∗ ca F nnbd + ( − j c + j d F nn ∗ db F ppac (cid:1) (cid:26) j a j c Kj d j b J (cid:27)(cid:19) . (20)The first two terms are for charge-changing transitions, while the last two are for charge-conserving transitions. Note it is possible to create an operator for just one direction, e.g.,a non-energy-weighted sum rule for β − transitions. B. Energy-weighted sum rules
We defineˆ O EWSR = 12 X M ( − M h ˆ F K, − M , [ ˆ H, ˆ F K,M ] i = X ab g ab [ j a ]( a † ⊗ ˜ b ) , + 14 X abcd ζ ab ζ cd X J W J ( ab, cd ) [ J ] (cid:16) A † J ( ab ) ⊗ ˜ A J ( cd ) (cid:17) , . (21)In this format the EWSR operator is an angular momentum scalar and, again, looks “justlike” a Hamiltonian, for purposes of use in a shell-model code.6n order to derive the EWSR as an expectation value of a double-commutator, we must use (5). Then, for example, for Gamow-Teller we cannot compute the EWSR for β − or β +alone, but must compute it for the sum. While this is physically less interesting, it is theonly possibility for an expectation value of a two-body operator. If we do not use (5), theEWSR becomes S ( E i ) = D i (cid:12)(cid:12)(cid:12) ˆ F † [ ˆ H, ˆ F ] (cid:12)(cid:12)(cid:12) i E = D i (cid:12)(cid:12)(cid:12) [ ˆ F † , ˆ H ] ˆ F (cid:12)(cid:12)(cid:12) i E = 12 D i (cid:12)(cid:12)(cid:12) ˆ F † [ ˆ H, ˆ F ] + [ ˆ F † , ˆ H ] ˆ F (cid:12)(cid:12)(cid:12) i E , (22)and the resulting operator will have three-body components.After annihilating commutators and recoupling angular momentums, the one-body partsof ˆ O EW SR in Eq.(21) are g ab = δ j a j b j a + 1) X cd ( − e ac F cd F ∗ bd + F ac e cd F ∗ bd + F ∗ ca e cd F db − F ∗ ca F cd e db ) , (23)where e ab are the one-body parts of the Hamiltonian in Eq.(11), and the two-body matrixelements of ˆ O EW SR are W J ( abcd ) = X i =1 W i ( abcd ; J ) , (24)with (using Eq. (14) where possible to eliminate or reduce phases) W ( abcd ; J ) = −
12 (1 + P cdJ ) X efJ ′ ( − J + J ′ (2 J ′ + 1) π J ′ de ζ ef ζ − cd V J ( ab, ef ) × F ec F fd (cid:26) J K J ′ j d j e j f (cid:27) (cid:26) J K J ′ j e j d j c (cid:27) , (25) W ( abcd ; J ) = −
12 (1 + P cdJ ) X efJ ′ (2 J ′ + 1) π J ′ cf ζ ce ζ − cd V J ( ab, ce ) × F ef F ∗ df (cid:26) J K J ′ j f j c j e (cid:27) (cid:26) J K J ′ j f j c j d (cid:27) , (26) W ( abcd ; J ) = (1 + P abJ )(1 + P cdJ ) X efJ ′ (2 J ′ + 1) ζ be ζ df ζ − ab ζ − cd V J ′ ( be, df ) × F ∗ ea F fc (cid:26) J K J ′ j e j b j a (cid:27) (cid:26) J K J ′ j f j d j c (cid:27) , (27) W ( abcd ; J ) = P ac P bd W ∗ ( abcd ; J ) , (28) W ( abcd ; J ) = P ac P bd W ∗ ( abcd ; J ) , (29)where ζ ab = √ δ ab as former defined, and π J ′ de is defined as π J ′ de = (cid:26) , if d = e and J ′ is odd;1 , else . (30)We introduce this symbol because in the derivations of W ( abcd ; J ), J ′ is an intermediateangular momentum, which accounts for the total angular momentum of two fermion anni-hilators in orbits d and e . As the Pauli principle demands, when d and e are the same orbit, J ′ must be even in (25). Similarly, in (26) when c and f are the same orbit, J ′ must beeven. For detailed explanations please see (A12-A13) and discussion there.7 II. RESULTS
Our formalism applies to configuration-interaction (CI) calculations in a shell-model basis.In CI calculations one diagonalizes the many-body Hamiltonian in a finite-dimensioned,orthonormal basis of Slater determinants, which are antisymmeterized products of single-particle wavefunctions, typically expressed in an occupation representation. The advantageof CI shell model calculations is that one can generate excited states easily, and for a modestdimensionality one can generate all the eigenstates in the model space.We use the
BIGSTICK
CI shell model code [38, 39] to calculate the many-body matrixelements H αβ = h α | ˆ H | β i and then solve ˆ H | i i = E i | i i . Greek letters ( α, β, . . . ) denote genericbasis states, while lowercase Latin letters ( i, j, . . . ) label eigenstates. As
BIGSTICK computesnot only energies but also wavefunctions, we can easily compute sum rules as an expecta-tion value, as in Eq. (6). We also tested our formalism by fully diagonalizing modest butnontrivial cases, with typical M -scheme dimensions on the order of a few thousand, wherewe compute transition density matrices and the subsequent transition strengths between allstates. This is a straightforward generalization of previous work on the NEWSR [7]. -2000200400 E W S R ( M e V e f m ) -4048 i (MeV)-40-20020406080 E W S R ( M e V e f m ) i (MeV) -20 E ce n t ( M e V ) (a) (b)(d)(c) FIG. 1. Energy weighted sum rules (EWSR) and transition strength function centroids as afunction of initial energy E i . Results are put into 2 MeV bins with the average and root-mean-square flucutation shown; the fluctuations are not sensitive to the size of the bins. (a) EWSRsfor isoscalar E2 for Cl in the sd shell. The (red) solid line is the secular behavior predicted byspectral distribution theory, as described in Ref. [7]. (b) Centroids for M1 transitions in Ne in the sd shell. (c) EWSR for E1 transitions in B in 0 p -1 s -0 d / space. (d) Centroids for Gamow-Tellertransitions, sum of β ± , for Ne in the sd shell. To illustrate our formalism we use phenomenological spaces and interactions, for example,the 1 s / -0 d / -0 d / or sd shell, using a universal sd interaction version ‘B’ (USDB) [40]. We8
10 12 14 16 18 20 A E W S R S M / E W S R T R K FIG. 2. Ground state E1 energy-weighted sum rule (EWSR) for Z = N nuclides computed in the0 p -1 s -0 d / shell model space (SM), normalized by the Thomas-Reiche-Kuhn (TRK) EWSR. show results for selected nuclides, for which we can fully diagonalize the Hamiltonian in themodel space, as a function of initial energy (relative to the ground state) in Fig. 1. The cen-troids are simply evaluated by the ratio of the EWSR to the NEWSR, as in Eq. (3). Becauseof the finite model space and because we consider the sum rules for all states, the centroidsand the EWSR must go from positive to negative. Panel (a) shows the EWSR for isoscalarE2 transitions in Cl, while panel (b) shows the centroids for transitions in Ne with stan-dard g -factors [41]. While we assume harmonic oscillator single-particle wave functions forthe basis, taking ~ Ω = 41 A − / MeV, because we compute centroids the oscillator lengthdivides out. All results were put into 2 MeV bins, but the size of the fluctuations shownby error bars are insensitive to the size of the bins. Also shown is the spectral distributiontheory prediction of the secular behavior: one exploits traces of many-body operators to ex-actly arrive at smooth secular behavior shown by the red solid line in panel 1(a). Not onlycan one compute the EWSR as an expectation value, the secular behavior with excitationenergy is quite smooth and by relating the EWSR to the expectation value of an operator,and defining an inner product using many-body traces, that behavior can be understoodfrom a simple mathematical point of view, as discussed in more detail in [7] (the reasonwe choose isoscalar E2 is that the publically available code we used to compute the innerproduct [42] only allows interactions with good isospin). Panel (d) shows the centroids forcharge-changing Gamow-Teller transitions starting from Ne. Because Eq. (6) requires thetransition operator of rank K to follow (5), we have to sum both β + and β − transitions.For Ne the total β − strength is 21.239 g A , which dominates over β + whose total strengthis 0.239 g A , satisfying the Ikeda sum rule. Again, because we are taking ratios the value of g A divides out for the centroids.We also considered E1 transitions in a space with opposite parity orbits, the 0 p / -0 p / -1 s / -0 d / or p - sd / space, chosen so we could fully diagonalize for some nontrivial cases.The interactions uses the Cohen-Kurath (CK) matrix elements in the 0 p shell[43], the olderUSD interaction [44] in the 0 d / -1 s / space, and the Millener-Kurath (MK) p - sd cross-shellmatrix elements[45]. Within the p and sd spaces the relative single-particle energies for the9K and USD interactions, respectively, are preserved, but sd single-particle energies shiftedrelative to the p -shell single particle energies to get the first 3 − state in O at approximately6 . O spectrum, in particular the first excited0 + state, is not very good, but the idea is to have a non-trivial model, not exact reproductionof the spectrum. Panel (c) of Fig. 1 shows the E1 EWSR for B, where, as with the othercases, due to the finite model space the sum rule is not constant. One of the most importantand most famous application of sum rules is to electric dipole (E1) transitions, where theThomas-Reiche-Kuhn (TRK) sum rule [24] predicts S = ( N Z/A ) e ~ / m N . Fig. (2) showsthe ground state E1 energy-weighted sum rule for Z = N nuclides in this space, normalizedby the TRK prediction. The enhancement over the TRK sum rule, between 40 and 125%, issimilar to previous results, [24, 46–50]. While one should not take these results as realistic,given the smallness of the model spaces and the crudity of the interaction, it nonethelessillustrates the simplicity of this approach.
10 12 14 16 18 N E ce n t ( E ) ( M e V ) NaAlPCl
10 12 14 16 18 2345 E x ( + ) ( M e V ) NeMgSiSAr (a) (b)
FIG. 3. In the sd -shell using the Brown-Richter USDB interaction. (a) Centroids of E2 transitionsfrom the ground state as a function of neutron number N . Includes nuclides with both even andodd proton number, with symbols in the boxes on both panels. (b) Excitation energies of the first2 + state for even-even nuclides only. By expressing sum rules as operators, one can efficiently search for systematic behaviors.For example, we searched for correlations in the sd shell suggested by Eq. (7) but found none.Further investigation instead led us to systematics of the E sd shell, shownin Fig. 3. Again we used the Brown-Richter USDB interaction, and used effective chargesof 1 . e and 0 . e for protons and neutrons, respectively. The USDB interaction is known tobe relatively good at producing low-lying energy spectra and transitions of sd -shell nuclei,so we use it to calculate E centroid ; while the E ~ Ω excitations,such transitions are excluded from this model space, so here the centroids mostly signal the10ow-lying transition strengths. The left panel, (a), gives the energy centroid, the ratio of theEWSR to the NEWSR easily calculated as expectation values, for isotopes of neon throughargon, for neutron numbe N = 9-19. The data suggest a convergence at the semi-magicclosure of the 0 d / shell at N = 14, which is a maximum for nuclides with Z <
14 and aminimum for
Z >
14. We have no simple explanation for this behavior, although it seemsclearly tied to the semi-magic nature of N = 14; it is quite different from the excitationenergy of the first 2 + energy in the even-even nuclides, shown in the right panel (b), which,although we do not show it, closely follow the experimental values. (The closest behaviorin the literature we can find are simple behaviors of 2 +1 and 4 +1 excitation energies in heavynuclei as a function of the number of valence protons and neutrons [51–54], demonstratingthe close relationship between collectivity and the proton-neutron interaction. However wefound that those simple relationships between the number of valence nucleons and the 2 +1 and 4 +1 energies do not hold in the sd shell.) We also note an advantage of sum rules overother regularities such as 2 + excitation energies: they can be applied easily to all nuclides,while E (2 +1 ) may signal the underlying structure of only even-even nuclei. Indeed Fig. 3(a)demonstrates this. Clearly much more exploration can be done. IV. SUMMARY
We presented explicit formulas of operators for non-energy-weighted ( S ) and energy-weighted ( S ) sums rules of transition strength functions, calculated as expectation values ina shell model occupation-space framework. These formulas are implemented in the publicallyavailable code PandasCommute [36], which can generate the sum rule operator one- and two-body matrix elements from general shell-model interactions and transition operator matrixelements. We presented examples of electromagnetic and weak transitions for typical casesin sd and psd / shell model spaces; sd shell calculations show that the centroids exhibit ansecular dependence on the parent state energy. Calculation of the E1 energy-weighted sumrule in a crude model space nonetheless show an enhancement over the Thomas-Reiche-Kuhnsum rule similar to previous results. We also showed intriguing systematics of E2 centroidsin the sd shell.This methodology can be further extended to no-core shell model spaces, even withisospin non-conserving forces (e.g. Coulomb force). As one only needs a parent state andthe Hamiltonian of the many-body system, E centroid might play the role of a test signal incalculations in sequentially enlarged spaces, thus may be useful to address e.g. quenching,impact of T = 0/ T = 1 interactions on strength functions and so on. Acknowledgement:
This material is based upon work supported by the U.S. Depart-ment of Energy, Office of Science, Office of Nuclear Physics, under Award Number DE-FG02-96ER40985, and National Natural Science Foundation of China (Grants No. 11225524 andNo. 11705100). This work is supported in part by the CUSTIPEN (China-U.S. TheoryInstitute for Physics with Exotic Nuclei)funded by the U.S. Department of Energy, Officeof Science under grant number de-sc0009971, which allowed C. W. Johnson to initiate thiscollaboration. Y. Lu is thankful for support to visit San Diego State University for 3 monthsand the hospitality extended to him, and with great pleasure thanks Prof. Y. M. Zhao for11seful discussions and most kind support.
Appendix A: Derivation of the double commutator
In this appendix we give some details of the derivation of the matrix elements for theEWSR operator, which requires double commutation. Given the one- and two-body matrixelements of the Hamiltonian, e ab and V J ( ab, cd ) as defined in (11), and the reduced one-body matrix elements F ab of the transition operator as in (13), we want to find the one-bodymatrix element g ab , and the two-body matrix elements W J ( ab, cd ) of the EWSR sum ruleoperator, as defined in (21). We remind the reader that we do not assume isospin symmetryand that the single-particle orbit labels, a, b, c, d , etc., may refer to distinct proton andneutron orbits.Taking the expression of the Hamiltonian in (12) into the double commutator in (21),ˆ O EW SR splits into two terms,ˆ O EWSR = −
12 ( − K [ K ] X ab e ab [ j a ] h [ ˆ Q ( ab ) , ˆ F K ] K , ˆ F K i −
18 ( − K [ K ] X abef ζ ab ζ ef X J V J ( ab, ef )[ J ] h [( A † J ( ab ) ⊗ ˜ A J ( ef )) , ˆ F K ] K , ˆ F K i (A1)where ˆ Q KM ( ab ) is defined as ˆ Q KM ( ab ) ≡ (ˆ a † ⊗ ˜ b ) KM . We have changed dummy indices inthe second term, so that V J ( ab, ef ) rather than V J ( ab, cd ) appears here, as it does in (25),for convenience of later explanations of how to derive (25).These terms involve commutators with angular momentum recouplings. Such commuta-tors are dealt with in a unified manner by authors of Ref. [55, 56] with a generalized Wicktheorem. We introduce their methodology in brief and return to (A1) with the borrowedtool. They define a generalized commutator,[ ˆ α, ˆ β ] = ˆ α ˆ β − θ αβ ˆ β ˆ α, (A2)where ˆ α, ˆ β are operators in occupation space, including single-particle fermion creation andannihilation operators, one-body transition operators, and fermion pair creation and anni-hilation operators. If j α , j β are the angular momenta of the operators, then θ αβ = ( − , j α , j β are half integers;1 , otherwise . (A3)With these definitions, it’s straight forward to derive[ ˆ α ˆ β, ˆ γ ] = ˆ α [ ˆ β, ˆ γ ] + θ βγ [ ˆ α, ˆ γ ] ˆ β. (A4)Now we also introduce a generalized commutator with good angular momentum coupling,[ ˆ α, ˆ β ] jm ≡ ( ˆ α ⊗ ˆ β ) jm − ( − j α + j β − j θ αβ ( ˆ β ⊗ ˆ α ) jm . (A5)12nd for spherical tensor products[( ˆ α ⊗ ˆ β ) j , ˆ γ ] j ′ = X j ′′ U ( j α j β j ′ j γ ; jj ′′ )( ˆ α ⊗ [ ˆ β, ˆ γ ] j ′′ ) j ′ + θ βγ X j ′′ ( − j α + j ′ − j − j ′′ U ( j α j β j γ j ′ ; jj ′′ )([ ˆ α, ˆ γ ] j ′′ ⊗ ˆ β ) j ′ , (A6)where U ( j α j β j γ j ′ ; jj ′′ ) ≡ ( − j α + j β + j γ + j ′ [ j ][ j ′′ ] ( j α j β jj ′ j γ j ′′ ) , (A7)and [ x ] ≡ √ x + 1 as defined before.Now we go back to (A1). We remind the reader that, according to (13), ˆ F K,M = P ab F ab [ K ] − ˆ Q K,M ( ab ), so the first term in (A1) is a linear summation of terms in theform of h [ ˆ Q ( ab ) , ˆ Q K ( cd )] K , ˆ Q K ( ef ) i .With (A6) we can derive h ˆ Q J ( ab ) , ˆ Q K ( cd ) i J ′ M ′ = h (ˆ a † ⊗ ˜ b ) J , (ˆ c † ⊗ ˜ d ) K i J ′ M ′ = ( − j a + j d + J ′ δ bc [ J ][ K ] ( j a j b JK J ′ j d ) ˆ Q J ′ M ′ ( ad ) − ( − j b + j c + J + K δ da [ J ][ K ] ( j a j b JJ ′ K j c ) ˆ Q J ′ M ′ ( cb ) , (A8)and thereafter hh ˆ Q J ( ab ) , ˆ Q K ( cd ) i J ′ , ˆ Q K ( ef ) i JM = [ J ][ J ′ ](2 K + 1) (cid:26) + φ aeK δ bc δ fa (cid:26) J K J ′ j d j a j b (cid:27) (cid:26) J K J ′ j a j d j e (cid:27) ˆ Q JM ( ed ) − φ dfJJ ′ δ bc δ de (cid:26) J K J ′ j d j a j b (cid:27) (cid:26) J K J ′ j d j a j f (cid:27) ˆ Q JM ( af )+ φ bfK δ ad δ be (cid:26) J K J ′ j c j b j a (cid:27) (cid:26) J K J ′ j b j c j f (cid:27) ˆ Q JM ( cf ) − φ ceJJ ′ δ ad δ cf (cid:26) J K J ′ j c j b j a (cid:27) (cid:26) J K J ′ j c j b j e (cid:27) ˆ Q JM ( eb ) (cid:27) , (A9)where φ aeK = ( − j a + j e + K , other φ ··· are similar. We take (A9) into the 1st term in (A1),and end up with the expression for g ab in (23). The second term in (A1) is a linear summation of terms h [( A † J ( ab ) ⊗ ˜ A J ( ef )) , ˆ F K ] K , ˆ F K i .With (A6) it’s straight forward to derive h(cid:16) A † J ( ab ) ⊗ ˜ A J ( ef ) (cid:17) , ˆ F K i K,M = X J ′ ( − J + K + J ′ [ J ′ ][ J ] − [ K ] − (cid:16) A † J ( ab ) ⊗ [ ˜ A J ( ef ) , ˆ F K ] J ′ (cid:17) K,M + X J ′ [ J ′ ][ J ] − [ K ] − (cid:16) [ A † J ( ab ) , ˆ F K ] J ′ ⊗ ˜ A J ( ef ) (cid:17) K,M , (A10)13nd thereafter h [( A † J ( ab ) ⊗ ˜ A J ( ef )) , ˆ F K ] K , ˆ F K i = X J ′ [ J ′ ][ J ] − [ K ] − n ( − J + K + J ′ (cid:16) A † J ( ab ) ⊗ h [ ˜ A J ( ef ) , ˆ F K ] J ′ , ˆ F K i J (cid:17) (A11)+2 (cid:16) [ A † J ( ab ) , ˆ F K ] J ′ ⊗ [ ˜ A J ( ef ) , ˆ F K ] J ′ (cid:17) + ( − J + K + J ′ (cid:16)h [ A † J ( ab ) , ˆ F K ] J ′ , ˆ F K i J ⊗ ˜ A J ( ef ) (cid:17) o . Linear summations of the 1st term in the brace of (A11) lead to W ( abcd ; J ) and W ( abcd ; J )in (25-26), the 2nd term to W ( abcd ; J ) in (27), and the 3rd term to W ( abcd ; J ) and W ( abcd ; J ) in (28-29). The symmetry between (25-26) and (28-29) originates from here.We take the 1st term in the brace of (A11) as an example, and explain restrictions causedby Pauli’s principle mentioned before. Use (A6) again to derive h ˜ A J ( ef ) , ˆ F K i J ′ M ′ = X gd F gd [ K ] − h ˜ A J ( ef ) , ˆ Q K ( gd ) i J ′ M ′ = − [ J ](1 + P efJ ) X d F fd ( j e j f JK J ′ j d ) ˜ A J ′ M ′ ( de ) . (A12)Based on (A12), we derive h ˜ A J ′ ( de ) , ˆ F K i JM and go further to h [ ˜ A J ( ef ) , ˆ F K ] J ′ , ˆ F K i J,M = X cdgh (2 K + 1) − F gd F hc h [ ˜ A J ( ef ) , ˆ Q K ( gd )] J ′ , ˆ Q K ( hc ) i JM = [ J ][ J ′ ](1 + P efJ ) X cd π J ′ de F fd F ec (cid:26) J K J ′ j d j e j f (cid:27) (cid:26) J K J ′ j e j d j c (cid:27) ˜ A JM ( cd ) (A13)+( − J + J ′ [ J ][ J ′ ](1 + P efJ ) X cd π J ′ de F fd F ∗ cd (cid:26) J K J ′ j d j e j f (cid:27) (cid:26) J K J ′ j d j e j c (cid:27) ˜ A JM ( ec ) . Note that ˜ A J ′ M ′ ( de ) does not show up in (A13), but as it appeared in (A12) as a necessarystone in the water, therefore the restriction by Pauli’s principle on ˜ A J ′ M ′ ( de ) is inheritedby (A13), i.e. when d and e in (25) are the same orbit J ′ must be even . So weintroduced π J ′ de as defined in (30) to stand for this restriction.We take the 1st term of (A13) into the 1st term in the brace of (A11), pick up factorsin (A1), and we end up with W ( abcd ; J ) in (25); similarly the 2nd term of (A13) end upwith W ( abcd ; J ) in (26). Naturally the restriction π J ′ de is inherited by W ( abcd ; J ) and also W ( abcd ; J ), but because we exchange indices when deriving W ( abcd ; J ), the restrictionbecomes π J ′ cf in (26).The same trick is applied to the other two terms in the brace of (A11), with (A6) it’sstraight forward to derive h ˆ A † J ( ab ) , ˆ F K i J ′ M ′ = X ef [ K ] − F ef h ˆ A † J ( ab ) , ˆ Q K ( ef ) i J ′ M ′ = ( − K [ J ](1 + P abJ ) X e F ∗ be (cid:26) J K J ′ j e j a j b (cid:27) ˆ A † J ′ M ′ ( ea ) , (A14)14nd thereafter h [ ˆ A † J ( ab ) , ˆ F K ] , ˆ F K i JM = [ J ][ J ′ ](1 + P abJ ) X eg π J ′ ae F ∗ be F ∗ ag (cid:26) J K J ′ j e j a j b (cid:27) (cid:26) J K J ′ j a j e j g (cid:27) ˆ A † JM ( ge )+( − J + J ′ [ J ][ J ′ ](1 + P abJ ) X eg π J ′ ae F ∗ be F ge (cid:26) J K J ′ j e j a j b (cid:27) (cid:26) J K J ′ j e j a j g (cid:27) ˆ A † JM ( ag ) . 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