aa r X i v : . [ nu c l - t h ] J a n , Partial conservation of seniority and nuclear isomerism
P. Van Isacker
Grand Acc´el´erateur National d’Ions Lourds,CEA/DSM–CNRS/IN2P3, BP 55027, F-14076 Caen Cedex 5, France
S. Heinze
Institute of Nuclear Physics, University of Cologne,Z¨ulpicherstrasse 77, 50937 Cologne, Germany (Dated: October 26, 2018)
Abstract
We point out the possibility of the partial conservation of the seniority quantum number whenmost eigenstates are mixed in seniority but some remain pure. This situation occurs in nucleifor the g / and h / shells where it is at the origin of the existence of seniority isomers in theruthenium and palladium isotopes. It also occurs for f bosons. PACS numbers: 03.65.Fd, 21.60.Cs, 21.60.Fw l n configuration where it appears as a label additional to the total orbital angularmomentum L , the total spin S , and the total angular momentum J [1]. About ten yearslater it was adopted in nuclear physics for the jj -coupling classification of nucleons in asingle j shell [2, 3]. These studies made clear the intuitive interpretation of seniority: itrefers to the number of particles that are not in pairs coupled to angular momentum J = 0.In nuclear physics the concept of seniority has proven extremely useful, especially in semi-magic nuclei where only one type of nucleon (neutron or proton) is active and where seniorityturns out to be conserved to a good approximation.Since the papers of Racah and Flowers appeared, a wealth of further results has beenobtained and it is by now well understood what are the necessary and sufficient conditionsfor an interaction to conserve seniority (see chapters 19 and 20 of Ref. [4]). To give a precisedefinition of these conditions, we introduce the following notations. We consider a systemof n particles with angular momentum j where for the sake of generality j can be integer forbosons or half-integer for fermions. A rotationally invariant two-body interaction ˆ V betweenthe particles is specified by its ⌊ j + 1 ⌋ matrix elements ν λ ≡ h j ; λ | ˆ V | j ; λ i (where ⌊ x ⌋ is thelargest integer smaller than or equal to x ). The notation | j ; λ i implies a normalized two-particle state with total angular momentum λ which can take the values λ = 0 , , . . . , p ,where 2 p = 2 j for bosons and 2 p = 2 j − V = P λ ν λ ˆ V λ where ˆ V λ is the operator defined via h j ; λ ′ | ˆ V λ | j ; λ ′′ i = δ λλ ′ δ λλ ′′ .With the above conventions the necessary and sufficient conditions for the conservationof seniority can be written as X λ a λjI ν λ = 0 , I = 2 , , . . . , p, (1)with a λjI √ λ + 1 = δ λI + 2 q (2 λ + 1)(2 I + 1) j j λj j I − " λ + 1)(2 I + 1)(2 j + 1)(2 j + σ )(2 j + 2 + σ )(2 j + 1 + 2 σ ) / , where the symbol between curly brackets is a Racah coefficient and σ ≡ ( − ) j is +1 forbosons and − ν λ by varying I between 2 and 2 p , it does not tell us how many of those areindependent. This number turns out to be ⌊ j/ ⌋ for bosons and ⌊ (2 j − / ⌋ for fermions,the number of independent seniority v = 3 states [7].Conservation of seniority does not, however, imply solvability. In general, even if aninteraction satisfies the conditions (1) and conserves seniority, that does not imply thatclosed algebraic expressions can be given for its eigenenergies and eigenfunctions. As regardsits characterization from the point of view of symmetries, seniority can be viewed as a partial dynamical symmetry. It is important to clarify first what exactly is meant by apartial dynamical symmetry which is an enlargement of the concept of dynamical symmetryas defined, e.g. , in chapter 11 of Ref. [8].The idea is to relax the conditions of complete solvability and this can be done in essentially two different ways:1. Some of the eigenstates keep all of the quantum numbers.
In this case the propertiesof solvability, good quantum numbers,and symmetry-dictated structure are fulfilledexactly, but only by a subset of eigenstates [9, 10].2.
All eigenstates keep some of the quantum numbers.
In this case none of the eigenstatesis solvable,yet some quantum numbers (of the conserved symmetries) are retained.Ingeneral, this type of partial dynamical symmetry arises if the hamiltonian preservessome of the quantum numbers in a dynamical-symmetry classification while breakingothers [11, 12].Combinations of 1 and 2 are possible as well, for example, if some of the eigenstates keepsome of the quantum numbers [13].How do seniority-conserving interactions fit in this classification? If the conditions (1)are satisfied by an interaction ˆ V , all its eigenstates carry the seniority quantum number v and, consequently, the second type of partial dynamical symmetry applies. The eigenstatesare not solvable in general but must be obtained from a numerical calculation. Nevertheless, some eigenstates are completely solvable for a general seniority-conserving interaction. Thiswas shown by Rowe and Rosensteel [5, 6] who derived closed, albeit complex, expressionsfor the energies of some multiplicity-free ( i.e. , unique for a given particle number n , angularmomentum J and seniority v ) n -particle states in a j = 9 / some eigenstates with good seniority. We recover anexample of this phenomenon which was pointed out earlier for the j = 9 / f bosons.To shed light on this problem of partial seniority conservation, we analyze the four-particlecase. The motivation for doing so is that the conditions (1) can be derived from the analysisof the three-particle case [4]. We might thus expect possible additional features to appearfor four particles which will indeed be confirmed by the analysis below.A four-particle state can be written as | j ( R ) j ( R ′ ); J i where two particles are first cou-pled to angular momentum R , the next two particles to R ′ and the intermediate angularmomenta R and R ′ to total J . This state is not (anti-)symmetric in all four particles andcan be made so by applying the (anti-)symmetry operator ˆ P , | j [ II ′ ] J i ∝ ˆ P | j ( I ) j ( I ′ ); J i = X RR ′ [ j ( R ) j ( R ′ ); J |} j [ II ′ ] J ] | j ( R ) j ( R ′ ); J i , where [ j ( R ) j ( R ′ ); J |} j [ II ′ ] J ] is a four-to-two-particle coefficient of fractional parentage(CFP). The square brackets [ II ′ ] label the four-particle state and indicate that it has beenobtained after (anti-)symmetrization of | j ( I ) j ( I ′ ); J i . The label [ II ′ ] defines an overcom-plete, non-orthogonal basis, that is, not all | j [ II ′ ] J i states with I, I ′ = 0 , , . . . , p areindependent. It is implicitly assumed that I and I ′ as well as R and R ′ are even.The four-to-two-particle CFPs are known in closed form in terms of 9 j symbols and,furthermore, the overlaps h j [ II ′ ] J | j [ LL ′ ] J i and the matrix elements h j [ II ′ ] J | ˆ V λ | j [ LL ′ ] J i can be expressed in terms of them. The expressions are rather cumbersome and are notgiven here but it is accepted that the overlaps and matrix elements are known as algebraicexpressions of the intermediate and final angular momenta.We assume in the following that J = 0, corresponding to four-particle states with seniority v = 2 or v = 4. By definition a seniority v = 2 four-particle state is | j , v = 2 , J i = | j [0 J ] J i . (2)4 seniority v = 4 state is constructed from | j [ II ′ ] J i with I, I ′ = 0 and it is orthogonal tothe state (2). It can thus be written as | j [ II ′ ] , v = 4 , J i = | j [ II ′ ] J i − h j [ II ′ ] J | j [0 J ] J i| j [0 J ] J i . (3)If there is more than one v = 4 state for a given J , the indices [ II ′ ] serve as an additionallabel. Seniority conservation of the interaction ˆ V implies h j , v = 2 , J | ˆ V | j [ II ′ ] , v = 4 , J i = 0 (4)or h j [0 J ] J | ˆ V | j [ II ′ ] J ih j [0 J ] J | ˆ V | j [0 J ] J i = h j [0 J ] J | j [ II ′ ] J i . (5)Insertion of the values for the four-to-two-particle CFPs yields the conditions (1).We now turn our attention to the problem of partial seniority conservation and derive theconditions for an interaction ˆ V to have some four-particle eigenstates with good seniority.Note that there are a number of ‘trivial’ examples of this. For example, if the total angularmomentum J is odd, a four-particle state cannot be of seniority v = 0 or v = 2 and mustnecessarily have seniority v = 4. Also, for J > p the four-particle state must be of seniority v = 4. These trivial cases are not of interest here. Instead, we study the situation whereboth v = 2 and v = 4 occur for the same J and where a general interaction ˆ V mixes the v = 2 state with a subset of the v = 4 states but not with all. A general seniority v = 4state is specified by the coefficients η II ′ in the expansion | j { η II ′ } , v = 4 , J i = X II ′ η II ′ | j [ II ′ ] , v = 4 , J i , (6)where the sum is over q linearly independent combinations [ II ′ ] (with I = 0 and I ′ = 0),as many as there are independent v = 4 states. Let us now focus on bosons with j ≤ j ≤ /
2. In these cases Eq. (1) yields only one condition and a generalinteraction can be written as a single component ˆ V λ plus an interaction that conservesseniority. Consequently, if the condition of partial seniority conservation is satisfied by asingle λ component, it will be valid for an arbitrary interaction. The fact that (6) is aneigenstate of ˆ V λ and that this interaction does not mix it with the v = 2 state is expressed5y X II ′ η II ′ h j [ LL ′ ] , v = 4 , J | ˆ V λ | j [ II ′ ] , v = 4 , J i = E λ X II ′ η II ′ h j [ LL ′ ] , v = 4 , J | j [ II ′ ] , v = 4 , J i , X II ′ η II ′ h j , v = 2 , J | ˆ V λ | j [ II ′ ] , v = 4 , J i = 0 . (7)There are q + 1 unknowns: the q coefficients η II ′ and the energy E λ . Equations (7) are also q + 1 in number and together with a normalization condition on the coefficients η II ′ theydefine an overcomplete set of equations in { η II ′ , E λ } not satisfied in general but possiblyfor special values of j and J . Furthermore, according to the preceding discussion, if theseequations are satisfied for one λ , they must be valid for all λ and in each case the solutionyields E λ , the eigenvalue of ˆ V λ . A symbolic solution of the Eqs. (7) (for general j and J )is difficult to obtain but, using the closed expressions for the overlaps and matrix elements,it is straightforward to find solutions for given j and J . In particular, a solution of theovercomplete set of equations is found for j = 9 / J = 4 ,
6. We thus confirm the findingof Refs. [14, 15] who noted the existence of these two states that have the distinctive propertyof having exact seniority v = 4 for any interaction ˆ V (barring accidental degeneracies).Solution of the Eqs. (7) for j = 9 / J = 4 , | (9 / , v = 4 , J = 4 i = s | (9 / [22] , v = 4 , J = 4 i − s | (9 / [24] , v = 4 , J = 4 i , | (9 / , v = 4 , J = 6 i = s | (9 / [24] , v = 4 , J = 6 i − s | (9 / [44] , v = 4 , J = 6 i . (8)These states are normalized but expressed in terms of basis states that are not orthonormal.In addition, the solutions E λ can be used to derive the following energy expressions: E [(9 / , v = 4 , J = 4] = 6833 ν + ν + 1315 ν + 11455 ν ,E [(9 / , v = 4 , J = 6] = 1911 ν + 1213 ν + ν + 336143 ν . These expressions give the absolute energies of the two states and are valid for an arbitrary interaction among j = 9 / Energy H MeV L Exp Theo Ru Exp Theo Pd + + + + + + + + + + + + + + + + + + + + + + + + + + FIG. 1: Experimental and calculated energy spectra of Ru and Pd. The Ru and Pd spectraare calculated with g / interactions derived from Mo and Cd, respectively, which are senioritybreaking. All levels up to 3 MeV are shown. The two solvable v = 4 states are indicated in thicklines. whether the interaction conserves seniority or not. Their excitation energies E x are notknown in closed form, however, since the J π = 0 + ground state is not solvable for a generalinteraction. In contrast, a generally valid result is the difference between the excitationenergies, which can be written as E x [(9 / , v = 4 , J = 6] − E x [(9 / , v = 4 , J = 4]= − E x [(9 / , J = 2] − E x [(9 / , J = 4]+ 215 E x [(9 / , J = 6] + 1865 E x [(9 / , J = 8] , associating the excitation energies of the J = 4 and 6, seniority v = 4 states in the four-particle system with those of the J = 2, 4, 6 and 8, seniority v = 2 states in the two-particlesystem.Another interaction-independent result that can be derived concerns transition matrixelements. For example, the electric quadrupole transition between the two states (8) ischaracterized by the B (E2) value B (E2; (9 / , v = 4 , J = 6 → (9 / , v = 4 , J = 4)= 209475176468 B (E2; (9 / , J = 2 → (9 / , J = 0) . This again defines a parameter-independent relation between a property of the two- andfour-particle systems. 7 .50.130.0880.035 1.821.19 0.671.08 0.27 u = u = u = u = u = u = u = + + + + + + + FIG. 2: E2 decay in the (9 / system as obtained with a seniority-conserving interaction. Thenumbers between the levels denote B (E2) values expressed in units of B (E2; 2 +1 → +1 ) of thetwo-particle system. There are several nuclear regions with valence neutrons or protons predominantly confinedto an orbit with j = 9 /
2, which can be the 1 g / or 1 h / shell. Of particular interest arethe nuclei Ru ( Z = 44) and Pd ( Z = 46) which have four proton particles or holes in the1 g / shell and a closed N = 50 configuration for the neutrons. The yrast J = 2 , , , v = 2 [16]. For anyreasonable interaction the solvable J = 4 , v = 2 states with the same J . This is illustrated in Fig. 1 which shows the observed yraststates in Ru and Pd and compares them with the levels calculated with two differentinteractions derived from Mo and Cd, respectively. For a constant interaction the Ruand Pd spectra (four particles and four holes in the g / shell) are identical. The differencebetween the calculated spectra in Fig. 1 gives an idea of the uncertainty on the energy whichmight be of use in the experimental search for the J π = 4 +2 , +2 states [17].Partial seniority conservation sheds also some new light on the existence of isomers asobserved in this region [18]. Figure 2 illustrates the E2 decay in the (9 / system asobtained with a seniority-conserving interaction. It displays a pattern of very small B (E2)values between v = 2 states which is typical of the seniority classification in nuclei near midshell ( n ≈ j + 1 /
2) and which is at the basis of the explanation of seniority isomers [16].The decay of the two solvable J = 4 , B (E2) values8hat are an order of magnitude larger. The results derived here imply that, in spite of beingclose in energy, the two solvable v = 4 states do not mix with the v = 2 states, even foran interaction that does not conserve seniority. Within a (9 / approximation, the patternshown in Fig. 2 is stable since any breaking of the seniority quantum number of the yrast J = 4 , v = 4 levels which lie morethan 1 MeV higher. Furthermore, the v = 4 components in the yrast states can be probedby detecting the M1 decay out of the solvable v = 4 states since the M1 operator cannotconnect components with different seniority.A search for solutions of Eqs. (7) did not reveal other cases of partial seniority conservationin fermionic systems with other j and/or J . However, numerical studies [19] have shown itsexistence in bosonic systems, in particular for f bosons, where we have been able to findanalytic energy expressions for several boson numbers, again valid for a general interaction.These findings suggest that the mechanism of partial seniority conservation with an arbitraryinteraction occurs in systems that are ‘only just’ not entirely solvable ( i.e. , j = 9 / j = 3 for bosons). This will the subject of future investigations.We wish to thank Larry Zamick, Alex Brown, Ami Leviatan, and Igal Talmi for illumi-nating discussions. [1] G. Racah,Phys. Rev. , 367 (1943).[2] G. Racah, in L. Farkas Memorial Volume , (Research council of Israel, Jerusalem, 1952) p. 294.[3] B.H. Flowers, Proc. Roy. Soc. (London) A , 248 (1952).[4] I. Talmi,
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