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The Interacting Boson Model (IBM) of the Atomic Nucleus,
an Introduction

Walter Pfeifer
172 pp, 35 line figures, 1998, revised 2008
Dr. Walter Pfeifer: Stapfenackerweg 9, CH 5034 Suhr, Switzerland

Contents

 

Preface

1  Introduction

2  Characteristics of the IBM

3  Many-body configurations

    3.1 Many-boson states

    3.2 Symmetric states of two and three d-bosons

    3.3 The seniority scheme, rules defining J

4  Many-boson states with undefined angular  momentum

    4.1 two- and three-d-boson states

    4.2 General "primitive" many-boson states

5  Operators and matrix elements

    5.1 Matrix elements of the single-boson operator

    5.2 Creation and annihilation operators

    5.3 Single- and two-boson operators represented by creation and annihilation operators

6  Applications of the creation and annihilation operators

    6.1 Many-boson configurations represented by operators

    6.2 Generating boson pairs with total angular momentum zero

    6.3 Tensor operators annihilating bosons

    6.4 Annihilating boson pairs with total angular momentum zero

    6.5 Number operators for bosons

7  The Hamilton operator of the IBM1

    7.1 The components of the Hamiltonian

    7.2 The operator of the boson-boson interaction formulated with defined angular momentum

    7.3 The basic form of the Hamilton operator

    7.4 The conservation of boson number

8  The angular momentum operator of the IBM1

    8.1 The angular momentum operator in quantum mechanics

    8.2 The angular momentum operator expressed in terms of d-boson operators

    8.3 The conservation of angular momentum

9  The Hamiltonian expressed in terms of Casimir operators

10  The u(5)- or vibrational limit

    10.1 The Hamiltonian of the vibrational limit. The spherical basis

    10.2 Eigenvalues of the seniority operator

    10.3 Energy eigenvalues. Comparison with experimental data

11  Electromagnetic transitions in the u(5)-limit

    11.1 Multipole radiation

    11.2 The operator of the electromagnetic interaction

    11.3 Transition probabilities

    11.4 Reduced matrix elements for çDnd ç = 1

    11.5 Comparison with experimental data of electric quadrupole transitions

    11.6 Transitions with çDnd ç= 0

    11.7 Configurations with nd  > t

    11.8 Quadrupole moments

12  The treatment of the complete Hamiltonian of  the IBM1

    12.1 Eigenstates

    12.2 Matrix elements of the Hamiltonian

    12.3 Electric quadrupole radiation

    12.4 Comparison with experimental data

    12.5 An empirical Hamilton operator

13  Lie algebras

    13.1 Definition

    13.2 The u(N)- and the su(N)-algebra

    13.3 The so(N) algebra

    13.4 Dimensions of three classical algebras

    13.5 Operators constituting Lie algebras, their basis functions and their Casimir operators

    13.6 Properties of operators

14  Group theoretical aspects of the IBM1

    14.1 Basis operators of the Lie algebras in the IBM1

    14.2 Subalgebras within the IBM1

    14.3 The spherical basis as function basis of Lie algebras

    14.4 Casimir operators of the u(5)- or vibrational limit

    14.5 Casimir operators of the su(3)- or rotational limit

    14.6 Casimir operators of the so(6)- or g -instable limit

15  The proton-neutron interacting boson model IBM2

    15.1 The complete Hamiltonian of the IBM2

    15.2 Basis states and the angular momentum operator of the IBM2

    15.3 A reduced Hamiltonian of the IBM2

    15.4 Eigenenergies and electromagnetic transitions within the IBM2

    15.5 Comparisons with experimental data

    15.6 Chains of Lie algebras in the IBM2

16  The interacting boson-fermion model IBFM

    16.1 The Hamiltonian of the IBFM

    16.2 The u(5)-limit of the IBFM

    16.3 Numerical treatment of the IBFM. Comparison with experimental data

Appendix

    A1 Clebsch-Gordan coefficients and 3-j symbols

    A2 Symmetry properties of coupled spin states of identical objects

    A3 Racah-coefficients and 6-j symbols

    A4 The 9-j symbol

    A5 The Wigner-Eckart theorem

    A6 A further multipole representation of the Hamiltonian

    A7 The program package PHINT

    A8 Commutators of operators such as [d +´d ~ ](J)M

    A9 The commutator [[d +´d ~ ](J�€™ )M�€™ , [d +´d + ](J)M]

References

Index

Preface

The interacting boson model (IBM) is suitable for describing intermediate and heavy atomic nuclei. Adjusting a small number of parameters, it reproduces the majority of the low-lying states of such nuclei. Figure 0.1 gives a survey of nuclei which have been handled with the model variant IBM2. Figures 10.7 and 14.3 show the nuclei for which IBM1-calculations have been performed.


 

The IBM is based on the well-known shell model and on geometrical collective models of the atomic nucleus. Despite its relatively simple structure, it has proved to be a powerful tool. In addition, it is of considerable theoretical interest since it shows the dynamical symmetries of several nuclei, which are made visible using Lie algebras.

The IBM was created in 1974 by F. Iachello and A. Arima ( Arima and Iachello, 1975 ). Subsequently in numerous papers it has been checked, extended, and discussed. In 1990 in Santa Fe, New Mexico, Akito Arima was awarded the Weatherhill medal by the Franklin institute for his many contributions to the field of nuclear physics. In the same year, Francesco Iachello received the Wigner medal given by the Group Theory and Fundamental Physics Foundation, which cited him for " developing powerful algebraic tools and models in nuclear physics ". In 1993, A. Arima and F. Iachello were awarded the T. W. Bonner Prize in Nuclear Physics by the American Physical Society.

The international symposia of Erice ( Italy, 1978 ), Granada ( Spain, 1981 ), Drexel ( USA, 1983 ), Gull Lake ( USA, 1984 ), La Rábida ( Spain, 1985 ), Dubrovnik ( Yugoslavia, 1986 ) as well as other events focusing on the IBM have clearly demonstrated the wide interest in this theory and its further development on an international scale. In 1994 in Padua ( Italy ), the "International Conference on Perspectives for the Interacting Boson Model on the Occasion of its 20th Anniversary" took place.

In recent years outstanding survey reports on the IBM have been published by Iachello and Arima ( 1987 ), Eisenberg and Greiner ( 1987 ), Talmi ( 1993 ), Frank and Van Isacker ( 1994 ) , and others. Unfortunately, there are few introductory books on the IBM available for the interested reader. The present publication might reduce this deficiency. It is directed towards physics students and experimental physicists interested in the main properties of the IBM. Knowledge of the elements of quantum mechanics, nuclear physics, and electrodynamics is a prerequisite.

The experienced reader may feel that some transformations and proofs have been rendered in too great detail. For the beginner, however, this is indispensable and may serve him as an exercise. 

1  Introduction

A model of the atomic nucleus has to be able to describe nuclear properties such as spins and energies of the lowest levels, decay probabilities for the emission of gamma quantas, probabilities ( spectroscopic factors ) of transfer reactions, multipole moments and so forth. In this chapter those models are outlined from which the IBM comes.

The IBM is mainly rooted in the shell model, which has proved to be an excellent instrument for light nuclei ( up to 50 nucleons ). The larger the number of nucleons becomes the more shells have to be taken into account and the number of nuclear states soon becomes so colossal that the shell model will be intractable. For example the 2+ state ( spin 2 and positive parity ) of 154Sm shows 3�—1014 different configurations ( Casten, 1990, p. 198 ). The interacting boson model ( sometimes named interacting boson approximation IBA ) reduces the number of states heavily. It constitutes only 26 configurations for the 2+ state mentioned above.

The shell model reveals that the low-lying states of the even-even nuclei are made up predominantly by nucleon pairs with total spin 0 or 2. Higher spins of such pairs are rare for energy reasons ( Hess, 1983, p. 55 ). Particularly the spins of pairs of identical nucleons are even numbers because they constitute an antisymmetric state ( appendix A2 ). Furthermore, in the case of two identical nucleon pairs the total spin is strictly even, which follows from the fact that the pairs behave like bosons ( see appendix A2 ). This theoretical result is not far from the real situation of even-even nuclei, from which it is known that their total spin predominantly is even.

These and other arguments led to the basic assumption of the IBM which postulates that the nucleon pairs are represented by bosons with angular momenta l = 0 or 2. The multitude of shells which appears in the shell model is reduced to the simple s-shell ( l = 0 ) and the d-shell ( l = 2 ) which is composed vectorially by d-bosons analogously to the shell model technique. The IBM builds on a closed shell i.e. the number of bosons depends on the number of active nucleon ( or hole ) pairs outside a closed shell. Each type of bosons, the s- and the d-boson, has its own binding energy with regard to the closed shell. Analogously to the standard shell model, the interacting potential of the bosons acts only in pairs.

as a peculiarity of the IBM there exist special cases in which certain linear combinations of matrix elements of this interaction potential vanish ( chapters 10 and 14 ). In these cases the energies of the nuclear states and the configurations can be expressed in a closed algebraic form. These special cases are named "dynamic symmetries". They correspond to the well-known "limits" allocated to the vibration, the rotation et cetera of the whole nucleus. However most nuclei have to be calculated by diagonalising the Hamilton matrix as is usual in quantum mechanics ( chapter 12 ).


The IBM is not only in connection with the shell model but also with the collective model of the atomic nucleus of Bohr and Mottelson ( 1953, 1975 ). In this model the deformation of the nuclear surface is represented by five parameters from which a Hamiltonian of a five dimensional oscillator results. It contains fivefold generating and annihilating operators for oscillator quanta. The operators of these bosons correspond to the operators of the d-shell in the IBM.

However, the handling of the collective model is laborious ( Jolos, 1985, p. 121 ). Moreover, the number of bosons is unlimited and is not a good quantum number in contrast to the situation in the IBM. The special cases mentioned above are reproduced by some versions of geometric models but they are not joined together continuously ( Barrett, 1981, p. 534 ). In the IBM these relations exist.

An additional relationship between both models consists in the fact that the form of the Hamilton operator ( after suitable transformations ) is similar to the one of the IBM ( Jolos, 1985, p. 124 ).

2  Characteristics of the IBM

The simplest versions of the IBM describe the even-even nucleus as an inert core combined with bosons which represent pairs of identical nucleons. Bosons behave symmetrically in the following way: supposing that each boson has a wave function, that can be attributed, the wave function of the total configuration does not alter if two bosons ( i.e. their variables) are interchanged. The analogy between nucleon pairs and bosons does not go so far that in the IBM the wave functions of the corresponding nucleons would appear. However, in the interacting boson-fermion model ( chapter 16 ) which deals with odd numbers of identical nucleons, bosons are coupled to nucleons. Bosons are taken as states without detailed structure and their symmetry properties result in commutation relations for the corresponding creation- and annihilation operators ( chapter 5 ).

The total spin of a boson is identical with its angular momentum i.e. one does not attribute an intrinsic spin to the bosons. Since the angular momenta of the bosons are even ( l = 0, 2 ) their parity is positive. Although plausible arguments exist for these angular momenta mentioned in the foregoing chapter, this choice is arbitrary and constitutes a typical characteristic of the theory ( however, exotic variants have been developed with l = 4 or odd values ). Only the success achieved by describing real nuclei justifies the assumption for the angular momenta.

The models IBM1 and IBM2 are restricted to nuclei with even numbers of protons and neutrons. In order to fix the number of bosons one takes into account that both types of nucleons constitute closed shells with particle numbers ..28, 50, 82 and 126 ( magic numbers ). Provided that the protons fill less than half of the furthest shell the number of the corresponding active protons has to be divided by two in order to obtain the boson number Np attributed to protons. If more than half of the shell is occupied the boson number reads Np = ( number of holes for protons )/2. By treating the neutrons in an analogous way, one obtains their number of bosons Nn . In the IBM1 the boson number N is calculated by adding the partial numbers i.e. N = Np + Nn . For example the nucleus  11854Xe64 shows the numbers Np = (54 - 50)/2 = 2, Nn = (64 -50)/2 = 7 and for 12854Xe74 the values Np = (54 - 50)/2 = 2, Nn = (82 - 74)/2 = 4 hold. Electromagnetic transitions don't alter the boson number but transfers of two identical nucleons lift or lower it by one.

Naturally the IBM has to take into account the fact that every nuclear state has a definite total nuclear angular momentum J or rather that the eigenvalue of the angular momentum operator J 2 is J(J + 1)�—h. J is an integer.


A boson interacts with the inert core of the nucleus ( having closed shells ) from which results its single boson energy e. Three-boson interactions are excluded in analogy with the assumptions of the standard shell model. In contrast to the collective model, in the IBM one does not obtain a semiclassical, vivid picture of the nucleus but one describes the algebraic structure of the Hamiltonian operator and of the states, for which reason it is named an algebraic model. 

3  Many-body configurations

At the beginning of this chapter the representation of boson configurations will be outlined and in the second section completely symmetric states of a few d-bosons will be formulated explicitly. In the end the rules are put together which hold for the collective states in the seniority scheme. They are compared with the results of section 3.2. In this chapter vector coupling technique is being applied, which is reviewed in the appendices A1 up to A3.

3.1 Many-boson states

Here we introduce a formulation of completely symmetric states of N bosons of which nd have a d-state and N - nd bosons are in the s-state. Besides the total angular momentum J and its projection M, for the most part additional ordering numbers are required in order to describe the collective state. One of these numbers is the seniority t, after which the most usual representation scheme is named.

For the moment we are leaving out the additional ordering number and write the completely symmetric configuration symbolically as

| (sN-nd)(0)0 (d nd)(J)M, J M ñ.  (3.1)

The s-bosons are coupled to a J = 0-state. In detail, the d-boson part is composed of single d-boson states having the angular momentum components 2, 1, 0, ‑1 and -2. These five single boson states dm appear in linear combinations as will be shown in the next section. The expression (3.1) is normalised to one.

3.2 Symmetric states of two and three d-bosons

In this section the s-bosons are left out of consideration and we will deal with the symmetrisation of configurations with a small number nd of d-bosons.

First we take nd = 2. According to the relation (A2.7) the configuration | d 2, J M ñ is symmetrical by itself if J is an even number. It has the form (A1.1)

| d 2, J M ñ = | åm1 m2 (2 m1 2 m2 | J M ) dm1 dm2 ñ º | [ d ´ d ](J)M ñ,   J = 0, 2, 4.  (3.2)

In order to obtain a three-boson state we couple one d-boson to a boson pair which has an even angular momentum J0 i.e. we form

| d 3, Jo J M ñ  º | [[ d ´ d ](J0) ´ d ](J)M ñ.  (3.3)

This expression is considered as a fully symmetrical three-d-boson state, which is obtained by carrying out a transposition procedure. In order to formulate this method, temporarily we are regarding bosons as distinguishable and we attribute an  individual number to each single boson  state. Supposing  that such
a state is described by a wave function, we have to label every variable with this boson number. We make use of the relation [ d(1)
´ d(2)](J0) = [ d(2) ´ d(1) ](J0), which holds for an even J0 according to ( A2.5 ) and ( A2.7 ). Starting from the partially symmetric ( p. s. ) form

| [ d(1) ´ d(2) ](J0) ´ d(3) ](J)M ñp.s.,

we obtain a symmetric three-boson state by adding two analogous forms in which the last d-boson is substituted as follows

A-1 | [[ d ´ d ](J0) ´ d ](J)M ñ = | [[ d(1) ´ d(2) ](J0) ´ d(3) ](J)M ñp.s.+

| [[ d(1) ´ d(3) ](J0) ´ d(2) ](J)M ñp.s. + | [[ d(3) ´ d(2) ](J0) ´ d(1) ](J)M ñp.s. .     (3.4)

A is the normalisation factor of the right hand side of (3.4). This expression is symmetric because one reproduces it by interchanging two boson numbers ( for example 2 and 3 ). In the last but one term, we can interchange d(1) and d(3) because it is partially symmetric. We employ the recoupling procedure (A3.3) and (A3.6) to the last two terms in (3.4) and obtain

 A-1 | [[ d ´ d ](J0) ´ d ](J)M ñ = | [[ d(1) ´ d(2) ](J0) ´ d(3) ](J)M ñp.s. +          (3.5)

åJ�€™ (-1)J�—�–(2J0 + 1)�—�–(2J�€™ + 1)�—{ 22  2J  J0J �€™}�—| [ d(3) ´ [ d(1) ´ d(2) ](J�€™ )](J)M ñp.s.+

åJ�€™ (-1)J�—�–(2J0 + 1)�—�–(2J�€™ + 1)�—{ 22  2J  J0J�€™}�—| [ d(3) ´ [ d(2) ´ d(1) ](J�€™ )](J)M ñp.s..

We now interchange d(2) and d(1) in the last term of (3.5), which yields the factor (-1)J�€™  ((A1.4)). both sums are added then, through which all terms with odd values J�€™ disappear. In the resulting sum we interchange d(3) and [ d(2) ´ d(1) ](J ), which annihilates the factor (-1)J according to (A1.4), because J�€™ is even. For formal reasons the first term on the right hand side of (3.5) is replaced by  åJ�€™even dJ�€™J0  | [[ d(1) ´ d(2) ](J�€™ ) ´ d(3) ](J)M ñp.s.. One obtains

A-1 | [[ d ´ d ](J0) ´ d ](J)M ñ =                                                                (3.6)

åJ'even (dJ�€™J0  + 2�—�–(2J0 + 1)�—�–(2J�€™ + 1)�—{ 22  2J  J0J �€™}) | [[ d(1) ´ d(2) ](J�€™ ) ´ d(3) ](J)M ñ.

The normalisation factor A reads

A = (3 + 6(2J0 + 1) { 22  2J  J0J0})-1/2.  (3.7)

It's a good exercise to derive this expression explicitly. The state | [d ´ d ](J0) ´ d ](J)M ñ is regarded as normalised to one. Analogously to the two-boson states (A1.9) here the partially symmetric states  |[[d(1)´d(2)](J�€™ )´d(3)](J)Mñ with different J�€™ are orthogonal to each other. We employ a slightly modified form of (3.5)

A-1 | [[ d ´ d ](J0) ´ d ](J)M ñ = | [[ d(1) ´ d(2) ](J0) ´ d(3) ](J)M ñp.s.+

�–(2J0 + 1)�—åJ�€™ (1 + (-1)J�€™)�–(2J�€™ + 1)�—{ 22  2J  J0J �€™}�—| [[ d(1) ´ d(2) ](J�€™ ) ´ d(3) ](J)M ñp.s. .

and make up the following equation
A-2
á[[d ´ d](J0) ´ d](J)M | [[ d ´ d ](J0) ´ d ](J)M ñ = A-2 =

1 + 2�—2�—(2J0 + 1){ 22  2J  J0J0} + (2J0 + 1) åJ�€™ (2 + 2�—(-1)J�€™)(2J�€™ + 1){ 22  2J  J0J �€™}2.

Due to (A3.8) and (A3.9) the relations

åJ�€™ (2J�€™ + 1) { 22  2J  J0J �€™}2 = (2J0 + 1)-1 and

åJ�€™ (-1)J�€™(2J�€™ + 1){ 22  2J  J0J �€™}2 = { 22  2J  J0J0} hold, from which we derive

A-2 = 1 + 2 + (2 + 4)(2J0 + 1){ 22  2J  J0J0}, which is in agreement with (3.7).

We now look into the J-values of symmetric three d-boson states represented in (3.6).

The case J = 0 is of some importance in the seniority scheme. The number of triplets with J = 0 is named nD i.e. in this case we have nD = 1.

For J ¹ 0 we insert the numerical values of the 6-j symbols (A3.12 - 14) in the equation (3.6). For J = 1 the partial vectors can only show J0 = J�€™ = 2 and the expression (3.6) vanishes. For J = 2 the values J0 = 0, 2, 4 have to be considered and the calculation yields

| [[ d ´ d ](0) ´ d ](2)M ñ = | [[ d ´ d ](2) ´ d ](2)M ñ = | [[ d ´ d ](4) ´ d ](2)M ñ .

We take a special interest in states with J0 = 0, that is why we treat the state (J0 = 2, J = 2) mentioned above as equivalent to (J0 = 4, J = 2) and to (J0 = 0, = 2) . Therefore we say, the configuration (J0 ¹ 0, J = 2) does not exist. In a similar way we see that the J = 3-states (J0 = 2, 4) differ only in their signs. Both J = 4-states (J0 = 2, 4) are identical. For J = 5 (J0 = 4) the expression (3.6) vanishes. J = 6 characterises the so-called "stretched" state.

3.3 The seniority scheme, rules defining J

General symmetric states of d-bosons are constructed by vector coupling and complete symmetrisation using group theory ( Hamermesh, 1962 ), ( Bayman and Landé, 1966 ). Here we have a look at the seniority scheme, which is the most common version of this representation. The configuration of nd d-bosons is written as follows

| nd , ([d ´ d](0)0)np�—([[d ´ d](2) ´ d](0)0)nD �—(dl)(J)M ñ .  (3.8)

In the expression (3.8) the doublet [d ´ d](0)0 with angular momentum 0 appears np  times and the triplet [[d ´ d](2) ´ d](0)0 exists nD times. The l remaining d-bosons constitute a configuration with the total angular momentum J (M) which contains neither a doublet nor a triplet with J = 0. Therefore the number of d-bosons reads nd = 2np + 3nD + l. The number t = nd - 2np = 3nD + l, which is left over after subtracting the doublets, is named seniority analogously to the description in the shell model. We name the configuration (d l)(J)M the "reduced" state of  the l bosons. It is defined unambiguously  by l, J  and  M (Talmi, 1993,
S. 763). Its total angular momentum J is identical with the one of the whole configuration (3.8).

In the seniority scheme the d-boson configurations are defined by the numbers nd, np, nD, J, (M).

There exist restrictions for the J-values. It can be shown that in a "reduced" state of l d-bosons the following values are permitted

J = l, l + 1, ... , 2l - 3, 2l - 2, 2l,  (3.9)

i. e. J < l and J = 2l - 1 are inadmissible. J = 2l represents the "stretched" state.

The exclusion of J = 2l - 1 in (3.9) can be explained in the following way. We know that for the "stretched", symmetric and to z orientated state of l d-bosons the relation J = M = 2l holds. We now construct the symmetric state with M = 2l - 1  and represent it using numbered bosons whose projections of the angular momentum is m :

            |(d l)M = 2l - 1 ñ = A |(d(1)m=1�—d(2)m=2�—d(3)m=2�—  ...    �—d(l)m=2

                                   +    d(1)m=2�—d(2)m=1�—d(3)m=2�—  ...     �—d(l)m=2  (3.10)

                                   +      .................................

                                   +    d(1)m=2�—d(2)m=2�—d(3)m=2�—   ...    �—d(l)m=1) ñ .

A is the normalisation constant. The expression (3.10) reveals that there exists only one state with M = 2l - 1. On the other hand, if one turns the "stretched" state ( with J = 2l ) relative to the z-axis in order to obtain the projection M = 2l - 1, the resulting state is still symmetric and must agree with the one of (3.10) because this is unique. For the same reason a state with J = 2l - 1 is not allowed because its maximal projection would be M = 2l - 1 which must not occur twice.

We now verify the rule (3.9) inspecting the boson states (3.2) and (3.6). For l = 2 the "reduced" state reads ç[ d ´ d ](J)M ñ with J ¹ 0. Owing to (3.9) only the values J = 2, 4 have to be considered which is in agreement with (3.2). For the "reduced" state with l = 3 according to (3.9) the values J = 0, 1, 2 are ruled out. In fact the discussion of equation (3.6) showed that J = 1 does not appear and that both other cases are equivalent to J0 = 0 which is inconsistent with the term "reduced" state. The rule (3.9) excludes J = 5 which has been found to be true for l = 3. Thus, for l = 2 and 3 the selection rule (3.9) is confirmed.

In table 3.1 for several boson numbers nd the allowed values np and nD are given. Accompanying values for t, l  and J are in the columns 3, 5 and 6.

Table 3.1. Classification of the d-boson configurations in the seniority scheme. nd : number of d-bosons,
n
p : number of boson pairs with total angular momentum 0,
n
D : number of boson triplets with total angular momentum 0,
t  :  seniority, l : number of bosons in the "reduced" state,
J :   total angular momentum

nd np t = nd - 2np nD  l = t -3nD      J
            
2 0 2 0 2 2,4
2 1 0 0 0 0
3 0 3 0 3 3,4,6
3 0 3 1 0 0
3 1 1 0 1 2
4 0 4 0 4 4,5,6,8
4 0 4 1 1 2
4 1 2 0 2 2,4
4 2 0 0 0 0
. . . . . .
. . . . . .
7 0 7 0 7 7,8,9,10,11,12,14
7 0 7 1 4 4,5,6,8
7 0 7 2 1 2
7 1 5 0 5 5,6,7,8,10
7 1 5 1 2 2,4
7 2 3 0 3 3,4,6
7 2 3 1 0 0
7 3 1 0 1 2
. . . . . .


Table 3.1 shows that for given nd > 3 some angular momenta J appear in more than one configuration. The value J = 1 is absent in the whole spectrum. Clearly it is missing also for l = 1 because this simplest "reduced" state consists of a single d-boson.

Among states with several d-bosons it happens that configurations with equal nd, t, J )-values differ in the quantity nD and are not orthogonal to one another. They have to be orthogonalised with the help of the well-known Schmidt procedure. By doing it, the number nD looses its character of an ordering number and it has to be replaced by an arbitrarily defined index.

Many-boson configurations in the seniority scheme stand out because they are eigenfunctions of the vibrational limit of the Hamilton operator ( chapter 10 and section 14.4 ). Since this special case correlates with the Lie algebra u(5) the states of the seniority scheme in addition are named u(5)-basis. "Spherical basis " is a further customary name. Besides this scheme there exist two less often used representations which are eigenfunctions of other limits of the Hamiltonian ( chapter 14 ).

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