A Derivation of the double commutator

# Transition sum rules in the shell model

## Abstract

An important characterization of electromagnetic and weak transitions in atomic nuclei are sum rules. 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 energy of transition strengths from an nuclear initial state to all allowed final states. These sum rules can be expressed as expectation values of operators, in the case of the EWSR a double commutator. While most prior applications of the double-commutator have been to special cases, we derive general formulas for matrix elements both operators in a shell model framework (occupation space), given the input matrix elements for the nuclear Hamiltonian and for the transition operator. With these new formulas, we easily evaluate centroids of transition strength functions, with no need to calculate daughter states. We apply this simple tool to a number of nuclides, and demonstrate the sum rules follow smooth secular behavior as a function of initial energy. We also find surprising systematic behaviors for ground state E2 centroids in the -shell.

## I Introduction

Atomic nuclei are neither static nor exist in isolation. Their transitions play important roles in fundamental, applied, and astro-physics, as well as revealing key information about nuclear structure beyond just excitation energies. In this paper we focus on electromagnetic and weak transitions; such transition strength distributions are important for -spectroscopy, nucleosynthesis and decays, as they are used to extract level densities Syed et al. (2009), calculate nuclear reaction rates in stellar processes Larsen and Goriely (2010) and analyze decay matrix elements Yako et al. (2009).

The strength function for a transition operator from an initial state at energy , to a final state at absolute energy and excitation energy is defined as

 S(Ei,Ex)=∑fδ(Ex−Ei+Ef)∣∣⟨f∣∣^F∣∣i⟩∣∣2 (1)

Sum rules are moments of the strength function,

 Sk(Ei)=∫(Ex)kS(Ei,Ex)dEx. (2)

Two of the most important sum rules, which we consider here, are , the non-energy-weighted sum rule (NEWSR) or total strength, and , the energy-weighted sum rule (EWSR). These sum rules provide compact information about strength functions. For example, the famous Ikeda sum rule Ikeda (1964) for Gamow-Teller transitions is the difference between the total strength and total strength:

 S0(GT−)−S0(GT+)=3(N−Z)g2A,

where is the axial vector coupling relative to the vector coupling . For investigations of ‘quenching’ of Osterfeld (1992), the NEWSR can be a probe of the missing strengths due to hypothesized cross-shell configurations.

The centroid of a strength distribution is just the ratio of the EWSR to the NEWSR,

 Ecentroid(Ei)=S1(Ei)S0(Ei). (3)

For a compact distribution of a giant resonance, is roughly the location of the resonance peak, relative to the parent state energy . Both the NEWSR and can be test signals of the validity of the general Brink-Axel hypothesis Guttormsen et al. (2016); Johnson (2015). The general Brink-Axel hypothesis Brink and Phil (1955); Axel (1962); Brink (2009) assumes that the strength distribution of transitions from any parent state is approximately the same, thus as a result is independent on . Though it seems this hypothesis needs to be modified for E1Ritman et al. (1993); Bracco et al. (1995); Angell et al. (2012), M1Larsen et al. (2007); Schwengner et al. (2013); Brown and Larsen (2014) (the low-energy anomaly) and GTMisch et al. (2014) transitions, it is still being widely used to calculate neutron-capture rates Tsoneva et al. (2015), extract nuclear level densities Schiller et al. (2003); Syed et al. (2009); Voinov et al. (2006) and can have a substantial impact on astrophysical relevance Goriely (1998); Larsen and Goriely (2010).

Sum rules are appealing not only because they characterize strength functions, but also because using closure some sum rules can be rewritten as expectation values of operators Ring and Schuck (2004). Allowing for transition operators with good angular momentum rank , one should sum over the -component , and the total strength becomes

 ∑f∑M|⟨f|^FK,M|i⟩|2=∑M⟨i|(^FK,M)†^FK,M|i⟩. (4)

Thus can be easily evaluated numerically without calculating any final state. The strength sum can be used to evaluate the former mentioned Ikeda Sum rule, useful as a check on computations.

The EWSR can be written as the expectation value of a double commutator, as long as the transition operator behaves as a spherical operator under Hermitian conjugation Edmonds (1996),

 (^FKM)†=(−1)M^FK,−M. (5)

The requirement of this will have consequences when we look at charge-changing transition such as decay. In that case, one must include both and transitions.

Invoking closure and Eq. (5), becomes

 ⟨i∣∣12∑M(−1)M[^FK,−M,[^H,^FK,M]]∣∣i⟩. (6)

As an example, the Thomas-Reiche-Kuhn sum rule evaluates the energy-weighted sum of strengths of an atom with electrons, and conserves to a constant proportional to . In nuclear physics the corresponding sum rule is similar, though the EWSR is proportional to because the dipole is relative to the center of mass. Another example is related with the “scissor mode” in rare-earth nuclei Heyde et al. (2010), for which the EWSR of low-lying ( MeV ) orbital M1 transitions shows a striking correlation with the transition,

 ∑fB(M1;0+1→1+f)E1+f∝∑fB(E2;0+1→2+f). (7)

This EWSR is derived both in the IBM-2 model Heyde and De Coster (1991), and in the shell model Zamick and Zheng (1992); Moya de Guerra and Zamick (1993) with phenomenological interactions.

In this paper we go beyond specific cases and, in the next section, write down the general form 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. Lu (2017) we make available a C++ code to generate those operator matrix elements. With such machinery one can directly compute the NEWSR and EWSR easily for many nuclides and many transitions. Prior work showed that the NEWSR follows simple secular behavior with the initial energy and gave a general argument Johnson (2015). In section III we show a few cases and also find simple secular behavior. We also look at systematics of ground state E2 centroids throughout the -shell and find unexpected behavior.

## Ii Formalism and formulas

We work in the configuration-interaction shell model, using the occupation representation Fetter et al. (2003) with fermion single-particle creation and annihilation operators , , respectively. As is standard, our operators have good angular momentum. The labels of each single-particle state include the magnitude of angular momentum and -component ; there are other important quantum numbers, in particular parity, orbital angular momentum and label for the radial wave function, but those values are absorbed into the values of matrix elements, so, for example, the details of our derivation are independent of whether or not one 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 ,, and ) and orbits, by which we mean the set of states with the same , but different . We assign fermion operators of different orbits different lower-case Latin letters: , , etc., to prevent a proliferation of subscripts. (In our derviations, when discussing generic operators, which may be single-fermion operators or composed of products and sums of operators, we use lower-case Greek letters: ) In order to make our results broadly usable, we will be slightly pedantic.

To denote generic operators coupled up to good total angular momentum and total -component , we use the notation . Hence we have the general pair creation operator

 ^A†JM(ab)=(^a†⊗^b†)JM (8)

with two particles in orbits and . We also introduce the adjoint of , the pair annihilation operator,

 ~AJM(cd)=−(~c⊗~d)JM (9)

Here we use the standard convention , where is the -component of angular momentum; this guarantees that if transforms as a spherical tensor, so does Edmonds (1996). An alternate notation is

 ^AJM(cd)=(^A†JM(cd))†=(−1)J+M~AJ,−M(cd) (10)

With this we can write down a standard form for any one- plus two-body Hamiltonian or Hamiltonian-like operator, which are angular momentum scalars. To simplify we use

 ^H=∑abeab^nab+14∑abcdζabζcd∑JVJ(ab,cd)∑M^A†JM(ab)^AJM(cd), (11)

where and . Here is the matrix element of the purely two-body part of between normalized two-body states with good angular momentum ; because is a scalar the value is independent of the -component . One can also write this, in slightly different formalism, as

 ∑abeab[ja](^a†⊗~b)0,0+14∑abcdζabζcd∑JVJ(ab,cd)[J](^A†J(ab)⊗~AJ(cd))0,0 (12)

where we use the notatation , which some authors write as (we use the former to avoid getting confused with operators which always are denoted by either or ).

Finally we also introduce one-body transition operators with good angular momentum rank and -component of angular momentum ,

 ^FK,M=∑abFab1[K](^a†⊗~b)K,M (13)

Here is the reduced one-body matrix element using the Wigner-Eckart theorem and the conventions of Edmonds Edmonds (1996). For non-charge-changing transitions, Eq. (5) implies

 Fab=(−1)ja−jbF∗ba. (14)

With these definitions and conventions, we can now work out general formulas for sum rules. An important issue will be isospin. Realistic operators, such as M1, strongly break isospin, and so rather than working in a formalism with good isospin we treat protons and neutrons as being in separate orbits. (Counter to this, we give one example with isoscalar E2 transitions in section III.)

### ii.1 Non-energy-weighted sum rules

The non-energy-weighted sum rule operator is given by

 ^ONEWSR=→F†⋅→F=∑M(FKM)†FKM=∑M(−1)MFK−MFKM. (15)

using Eq. (5). Then

By writing out the operator as an angular momentum scalar and to look “just like” a Hamiltonian, for purposes of use in a shell-model code, we have

 ^ONEWSR= ∑abgab[ja](a†⊗~b)0,0+14∑abcdJζabζcdWJ(ab,cd)[J][A†J(ab)⊗~AJ(cd)]0,0,

where the single-particle matrix element is

 gab=∑cF∗caFcb2ja+1. (18)

We do not assume isospin symmetry, but assume our orbital labels also reference protons/neutrons. So in (18) labels and must be the same, proton or neutron. Now for the two-body matrix elements: for identical particles in orbits (i.e., all label protons or all label neutrons), we need to enforce antisymmetry, that is, , etc:

where , and is the exchange operator swapping . Here the only terms in which contribute are the non-charge-changing pieces, and .

For proton-neutron interactions, where we assume labels are proton and are neutron, i.e., we want to compute , we need to identify the proton-neutron parts of . So we still have (18) and

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 transitions.

### ii.2 Energy-weighted sum rules

We define

 ^OEWSR = 12∑M(−1)M[^FK,−M,[^H,^FK,M]] = ∑abgab[ja](a†⊗~b)0,0+14∑abcdζabζcd∑JWJ(ab,cd)[J][A†J(ab)⊗~AJ(cd)]0,0,

In this format the EWSR operator is an angular momentum scalar and, again, looks “just like” a Hamiltonian, for purposes of use in a shell-model code.

In order to derive the EWSR as a double-commutator, we must use (5). Then, for example, for Gamow-Teller we cannot compute the EWSR for or alone, but must compute it for the sum. While this is physically less interesting, it is the only possibility for an expectation value of a two-body operator. If one does not assume (5), the resulting operator will have three-body components.

After annihilating commutators and recoupling angular momentums, the one-body parts of in Eq.(II.2) are

 gab=δjajb2(2ja+1)∑cd(−eacFcdF∗bd+FacecdF∗bd+F∗caecdFdb−F∗caFcdedb), (22)

where are the one-body parts of the Hamiltonian in Eq.(11), and the two-body matrix elements of are

 WJ(abcd)=5∑i=1Wi(abcd;J), (23)

with (using Eq. (14) where possible to eliminate or reduce phases)

 W1(abcd;J) = Missing or unrecognized delimiter for \left W2(abcd;J) = −12(1+PcdJ)∑efJ′(2J′+1)ζceζ−1cdVJ(ab,ce)FefF∗df{JKJ′jfjcje}{JKJ′jfjcjd}, W3(abcd;J) = 2(1+PabJ)∑efJ′(2J′+1)ζbeζdfζ−1abζ−1cdVJ′(be,df)F∗eaFfc{JKJ′jejbja}{JKJ′jfjdjc}, W4(abcd;J) = PacPbdW1∗(abcd;J), W5(abcd;J) = PacPbdW2∗(abcd;J), (24)

where as former defined.

## Iii 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 advantage of CI shell model calculations is that one can generate excited states easily, and for a modest dimensionality one can generate all the eigenstates in the model space.

We use the BIGSTICK CI shell model code Johnson et al. (2013) to calculate the many-body matrix elements and then solve Greek letters () denote generic basis states, while lowercase Latin letters () label eigenstates. As BIGSTICK computes not only energies but also wavefunctions, we can easily compute sum rules as an expectation value, as in Eq. (6). We also tested our formalism by fully diagonalizing modest but nontrivial cases, with typical -scheme dimensions on the order of a few thousand, where we compute transition density matrices and the subsequent transition strengths between all states. This is a straightforward generalization of previous work on the NEWSR Johnson (2015).

To illustrate our formalism we use phenomenological spaces and interactions, for example, the -- or shell, using a universal interaction version ‘B’ (USDB) Brown and Richter (2006). We show results for selected nuclides, for which we can fully diagonalize the Hamiltonian in the model space, as a function of initial energy (relative to the ground state) in Fig. 1. The centroids are simply evaluated by the ratio of the EWSR to the NEWSR, as in Eq. (3). Because of the finite model space and because we consider the sum rules for all states, the centroids and the EWSR must go from positive to negative. Panel (a) shows the EWSR for isoscalar E2 transitions in Cl, while panel (b) shows the centroids for transitions in Ne with standard -factors Brussard and Glaudemans (1977). While we assume harmonic oscillator single-particle wave functions for the basis, taking MeV, because we compute centroids the oscillator length divides out. All results were put into 2 MeV bins, but the size of the fluctuations shown by errors bars are insensitive to the size of the bins. Also shown is the spectral distribution theory prediction of the secular behavior, as discussed in more detail in Johnson (2015) (the reason we choose isoscalar E2 is that the publically available code we used to compute the inner product Launey et al. (2014) only allows interactions with good isospin). The takeaway from this plot is not only can one compute the EWSR as an expectation value, the secular behavior with excitation energy is quite smooth and can be understood from a simple mathematical point of view. Panel (d) shows the centroids for charge-changing Gamow-Teller transitions starting from Ne. Because Eq. (6) requires the transition operator of rank to follow (5), we have to sum both and transitions. For Ne the total strength is 21.239 , which dominates over whose total strength is 0.239 , satisfying the Ikeda sum rule. Again, because we are taking ratios the value of divides out for the centroids.

We also considered E1 transitions in a space with opposite parity orbits, the --- or - space, chosen so we could fully diagonalize for some nontrivial cases. The interactions uses the Cohen-Kurath (CK) matrix elements in the shellCohen and Kurath (1965), the older USD interaction Wildenthal (1984) in the - space, and the Millener-Kurath (MK) - cross-shell matrix elementsMillener and Kurath (1975). Within the and spaces the relative single-particle energies for the CK and USD interactions, respectively, are preserved, but single-particle energies shifted relative to the -shell single particle energies to get the first state at approximately MeV above the ground state. The rest of the spectrum, in particular the first excited state, is not very good, but the idea is to have a non-trivial model, not exact reproduction of the spectrum. Panel (c) of Fig. 1 shows the E1 EWSR for B, where, as with the other cases, due to the finite model space the sum rule is not constant. Although one of the most important and most famous application of sum rules is to electric dipole (E1) transitions, where the Thomas-Reiche-Kuhn sum rule predicts , our model space is too small and the interaction too crude to satisfactorily test this.

By expressing sum rules as operators, one can efficiently search for systematic behaviors. For example, we searched for correlations in the shell suggested by Eq. (7) but found none. Further investigation instead led us to systematics of the transitions in the shell, shown in Fig. 2. Again we used the Brown-Richter USDB interaction, and used effective charges of and for protons and neutrons, respectively. The left panel, (a), gives the energy centroid, the ratio of the EWSR to the NEWSR easily calculated as expectation values, for isotopes of neon through argon, for neutron number -. The data suggest a convergence at the semi-magic closure of the shell at , which is approximately a maximum for nuclides with and a minimum for . We have no simple explanation for this behavior, although it seems clearly tied to the semi-magic nature of ; it is quite different from the excitation energy of the first 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 behavior in the literature we can find are simple behaviors of and excitation energies in heavy nuclei as a function of the number of valence protons and neutrons Casten (1985); Casten et al. (1987); Rangacharyulu et al. (1991); Brenner et al. (1992), demonstrating the close relationship between collectivity and the proton-neutron interaction. However we found that those simple relationships between the number of valence nucleons and the and energies do not hold in the shell.) We also note an advantage of sum rules over other regularities such as excitation energies: they can be applied easily to all nuclides, while may signal the underlying structure of only even-even nuclei. Indeed Fig. 2(a) demonstrates this. Clearly much more exploration can be done.

## Iv Summary

We presented explicit formulas of operators for non-energy-weighted () and energy-weighted () sums rules of transition strength functions, calculated as expectation values in a shell model occupation-space framework. These formulas are implemented in the publically available code PandasCommute Lu (2017), which can generate the sum rule operator one- and two-body matrix elements from general shell-model interactions and transition operator matrix elements. We presented examples of electromagnetic and weak transitions for typical cases in and shell model spaces; shell calculations show that the centroids exhibit an secular dependence on the parent state energy. We also showed intriguing systematics of E2 centroids in the shell.

This methodology can be further extended to no-core shell model spaces, even with isospin non-conserving forces (e.g. Coulomb force). As one only needs a parent state and the Hamiltonian of the many-body system, might play the role of a test signal in calculations in sequentially enlarged spaces, thus may be useful to address e.g. quenching, impact of / interactions on strength functions and so on.

Acknowledgement: This material is based upon work supported by the U.S. Department 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 and No. 11705100). This work is supported in part by the CUSTIPEN (China?U.S. Theory Institute for Physics with Exotic Nuclei) funded by the U.S. Department of Energy, Office of Science under grant number DE-SC0009971, which allowed C. W. Johnson to initiate this collaboration. Y. Lu is thankful for support to visit San Diego State University for 3 months and the hospitality extended to him, and with great pleasure thanks Prof. Y. M. Zhao for useful 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 the EWSR operator, which requires double commutation. Given the one- and two-body matrix elements of the Hamiltonian, and as defined in (11), and the reduced one-body matrix elements of the transition operator as in (13), we want to find the one-body matrix element , and the two-body matrix elements of the EWSR sum rule operator, as defined in (LABEL:Od.c.expansion). We remind the reader that we do not assume isospin symmetry and that the single-particle orbit labels, , etc., may refer to distinct proton and neutron orbits.

To derive the matrix elements, we use general formulas of commutators with angular momentum recouplings Chen et al. (1993); Chen (1993), in particular a generalized Wick theorem Chen (1993) starting from a generalized commutator,

 [^α,^β]=^α^β−θαβ^β^α, (25)

where are operators in occupation space, including single-particle fermion creation and annihilation operators, one-body transition operators, and fermion pair creation and annihilation operators. If are the angular momenta of the operators, then

 θαβ={−1,jα,jβ are half integers;1,otherwise. (26)

With these definitions, it’s straight forward to derive

 [^α^β,^γ]=^α[^β,^γ]+θβγ[^α,^γ]^β. (27)

Now we also introduce a generalized commutator with good angular momentum coupling,

 [^α,^β]jm≡(^α⊗^β)jm−θαβ(^β⊗^α)jm. (28)

and for spherical tensor products

 [(^α⊗^β)j,^γ]j′=∑j′′U(jαjβj′jγ;jj′′)(^α⊗[^β,^γ]j′′)j′ +θβγ∑j′′(−1)jα+j′−j−j′′U(jαjβjγj′;jj′′)[(^α,^γ)j′′⊗^β]d, (29)

where

 Unknown environment '% (30)

and as defined before. Eq.(29) breeds useful commutators as needed. For convenience we define

 ^QKM(ef)=(^e†⊗~f)KM, (31)

as a general one-body operator hence . By applying Eq.(29) twice, we have

 [^A†J(ab),^QK(ef)]J′M′=[(^a†⊗^b†)J,(^e†⊗~f)K]J′,M′ Missing or unrecognized delimiter for \left (32)

where , and is the exchange operator swapping , as defined before. Applying Eq. (32) twice we get

 {[^A†J(ab),^QK(ef)]J′,^QK(gh)}JM =(1+PabJ)(−1)je+jg−jf−jhδbf[J][J′](2K+1){ja  jb  JK  J′  je} Unknown environment '% (33)

Similarly we have

 [~AI(cd),^QK(ef)]J′M′=−[(~c⊗~d)I,(^e†⊗~f)K]J′M′ =−(1+PcdJ)δde[K][J]{jc  jd  JK  J′ jf}~AJ′M′(fc), (34)

and

 [[~AI(cd),^QK(ef)]J′,^QK(gh)]JM =(1+PcdJ)δde(2K+1)[J][J′]{jc  jd  JK  J′  jf}(1+PcfJ′)δcg{jf  jc  J′K  J  jh}~AJM(hf).

For commutators among one-body operators,

 [^QJ(ab),^QK(cd)]J′M′=[(^a†⊗~b)J,(^c†⊗~d)K]J′M′ Unknown environment '% −(−1)jb+jc+J+Kδda[J][K]{ja  jb  JJ′  K  jc}^QJ′M′(cb), (36)

and

 [[^QJ(ab),^QK(ef)]J′,QK(gh)]IM=[J][J′](2K+1) {−ϕabghKδbeδha{JKJ′jfjajb}{JKJ′jhjfjg}^QJM(gf) Unknown environment '% Unknown environment '% +ϕbhKδfaδbg{JKJ′jejbja}{JKJ′jhjejg}^QJM(eh)},

where , other are similar.

Taking Eq.(11) into Eq.(II.2), we see that is a linear summation of expressions; for one-body parts; and

 {[[^A†I,^QK]J′,^QK]J⊗~AJ]0,(^A†J⊗[[~AJ,^QK]J′,^QK]J)0,([^A†J,^QK]J′⊗[~AJ,^QK]J′)0

for two-body parts. Take Eq.(32-LABEL:[(Q,Q),Q]) into these forms, tidy up and we get and in Eq. (22 -24).

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