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

an Introduction

Walter Pfeifer

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 ç*Dn _{d}*

_{ }ç = 1

11.5 Comparison with experimental data of electric quadrupole transitions

11.6 Transitions
with ç*Dn _{d }*ç= 0

11.7 Configurations
with *n _{d}*

_{ }>

*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

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

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 ^{154}Sm shows 3�—10^{14} 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

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

### 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 *N** _{p}*
attributed to protons. If more than half of the shell is occupied the boson
number reads

*N*

*= ( number of holes for protons )/2. By treating the neutrons in an analogous way, one obtains their number of bosons*

_{p}*N*

*. In the IBM1 the boson number*

_{n}*N*is calculated by adding the partial numbers i.e.

*N*=

*N*

*+*

_{p}*N*

*. For example the nucleus*

_{n }^{118}

_{54}Xe

_{64}shows the numbers

*N*

*= (54 - 50)/2 = 2,*

_{p}*N*

*= (64 -50)/2 = 7 and for*

_{n}^{128}

_{54}Xe

_{74}the values

*N*

*= (54 - 50)/2 = 2,*

_{p }*N*

*= (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.*

_{n}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 *n _{d}*

*have a*

_{ }*d*-state and

*N*-

*n*bosons are in the

_{d}*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

| (*s ^{N-n}*

*)*

^{d}^{(}

^{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 *d** _{m} *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* n _{d}*
of

*d*-bosons.

First
we take *n _{d}*
= 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 = | å

_{m}

_{1}_{ }

_{m}*(2*

_{2}*m*2

_{1}*m*|

_{2}*J*

*M*)

*d*

_{m}

_{1}*d*

_{m}*ñ º | [*

_{2}*d*´

*d*]

^{(J)}

*ñ,*

_{M}*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 *J _{0}* i.e. we form

| *d ^{ }*

^{3},

*J*

_{o}

*J*

*M*ñ º | [[

*d*´

*d*]

^{(J}

^{0}^{)}´

*d*]

^{(J)}

*ñ. (3.3)*

_{M}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)]^{(J}^{0)} = [ *d*(2) ´ *d*(1)
]^{(J}^{0)}, which holds for an even *J*_{0} according to ( A2.5 ) and
( A2.7 ). Starting from the partially symmetric ( p.
s. ) form

| [ *d*(1) ´ *d*(2)
]^{(J}^{0}^{)} ´ *d*(3)
]^{(J)}* _{M}* ñ

_{p.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*
]^{(J}^{0}^{)} ´ *d*
]^{(J)}* _{M}* ñ = | [[

*d*(1) ´

*d*(2) ]

^{(J}

^{0}^{)}´

*d*(3) ]

^{(J)}

*ñ*

_{M}_{p.s.}+

| [[ *d*(1) ´ *d*(3) ]^{(J}^{0}^{)} ´ *d*(2)
]^{(J)}* _{M}* ñ

_{p.s.}+ | [[

*d*(3) ´

*d*(2) ]

^{(J}

^{0}^{)}´

*d*(1) ]

^{(J)}

*ñ*

_{M}_{p.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*
]^{(J}^{0}^{)} ´ *d*
]^{(J)}* _{M}* ñ = | [[

*d*(1) ´

*d*(2) ]

^{(J}

^{0}^{)}´

*d*(3) ]

^{(J)}

*ñ*

_{M}_{p.s.}+ (3.5)

å* _{J�€™ }*(-1)

*�—�–(2*

^{J}*J*+ 1)�—�–(2

_{0}*J�€™*+ 1)�—{

^{ 2}

_{2}

^{2}

_{J}

^{J}

^{0}_{J }**}�—| [**

_{�€™}*d*(3) ´ [

*d*(1) ´

*d*(2) ]

^{(J�€™ )}]

^{(J)}

*ñ*

_{M}_{p.s.}+

å* _{J�€™ }*(-1)

*�—�–(2*

^{J}*J*+ 1)�—�–(2

_{0}*J�€™*+ 1)�—{

^{ 2}

_{2}

^{2}

_{J}

^{J}^{0}

*}�—| [*

_{J�€™}*d*(3) ´ [

*d*(2) ´

*d*(1) ]

^{(J�€™ )}]

^{(J)}

*ñ*

_{M}_{p.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)

*according to (A1.4), because*

^{J}*J�€™*is even. For formal reasons the first term on the right hand side of (3.5) is replaced by å

_{J�€™}_{even}

*d*

_{J�€™J}

_{0}*| [[*

_{ }*d*(1) ´

*d*(2) ]

^{(J�€™ )}´

*d*(3) ]

^{(J)}

*ñ*

_{M}_{p.s.}. One obtains

*A*^{-1} | [[ *d*
´ *d* ]^{(J}^{0}^{)} ´ *d* ]^{(J)}* _{M}* ñ = (3.6)

å_{J'}_{even} (*d _{J�€™J}*

_{0}*+ 2�—�–(2*

*J*+ 1)�—�–(2

_{0}*J�€™*+ 1)�—{

^{ 2}

_{2}

^{2}

_{J}

^{J}

^{0}_{J }**})**

_{�€™}*| [[*

_{ }*d*(1) ´

*d*(2) ]

^{(J�€™ )}´

*d*(3) ]

^{(J)}

*ñ.*

_{M}The
normalisation factor *A*
reads

*A* = (3 + 6(2*J _{0}* + 1) {

^{ 2}

_{2}

^{2}

_{J}

^{J}

^{0}_{J}*})*

_{0}^{-1/2}. (3.7)

It's a
good exercise to derive this expression explicitly. The state | [[ *d* ´ *d* ]^{(J}^{0}^{)} ´ *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)}

*ñ with different*

_{M}*J�€™*are orthogonal to each other. We employ a slightly modified form of (3.5)

*A*^{-1}
| [[ *d* ´ *d*
]^{(J}^{0}^{)} ´ *d*
]^{(J)}* _{M}* ñ = | [[

*d*(1) ´

*d*(2) ]

^{(J}

^{0}^{)}´

*d*(3) ]

^{(J)}

*ñ*

_{M}_{p.s.}+

�–(2*J _{0}* + 1)�—å

*(1 + (-1)*

_{J�€™ }*)�–(2*

^{J�€™}*J�€™*+ 1)�—{

^{ 2}

_{2}

^{2}

_{J}

^{J}

^{0}_{J }**}�—|**

_{�€™}^{ }[[

*d*(1) ´

*d*(2) ]

^{(J�€™ )}´

*d*(3) ]

^{(J)}

_{M}^{ }ñ

_{p.s.}

_{ }.

and
make up the following equation

*A*^{-2 }á[[*d*
´ *d*]^{(J}^{0}^{)} ´ d]^{(J)}_{M} | [[ *d*
´ *d* ]^{(J}^{0}^{)} ´ *d* ]^{(J)}* _{M}* ñ =

*A*

^{-2}=

1 + 2�—2�—(2*J**0* + 1){^{ 2}_{2} ^{2}_{J}^{J}^{0}_{J}* _{0}*} + (2

*J*

*0*+ 1) å

*(2 + 2�—(-1)*

_{J�€™ }*)(2*

^{J�€™}*J�€™*+ 1){

^{ 2}

_{2}

^{2}

_{J}

^{J}

^{0}_{J }**}**

_{�€™}^{2}.

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

å* _{J�€™ }*(2

*J�€™*+ 1) {

^{ 2}

_{2}

^{2}

_{J}

^{J}

^{0}_{J }**}**

_{�€™}^{2}= (2

*J*

*0*+ 1)

^{-1}and

å* _{J�€™ }*(-1)

*(2*

^{J�€™}*J�€™*+ 1){

^{ 2}

_{2}

^{2}

_{J}

^{J}

^{0}_{J }**}**

_{�€™}^{2}= {

^{ 2}

_{2}

^{2}

_{J}

^{J}

^{0}_{J}*} hold, from which we derive*

_{0}
*A*^{-2}
= 1 + 2 + (2 + 4)(2*J**0* + 1){^{ 2}_{2} ^{2}_{J}^{J}^{0}_{J}* _{0}*}, 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 *n** _{D}* i.e.
in this case we have

*n*

*= 1.*

_{D}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 *J**0* = *J�€™* = 2 and the expression (3.6) vanishes. For *J* = 2 the values *J**0* = 0, 2, 4 have to be
considered and the calculation yields

| [[ *d* ´ *d* ]^{(0)} ´ *d* ]^{(2)}* _{M}* ñ = | [[

*d*´

*d*]

^{(2)}´

*d*]

^{(2)}

*ñ = | [[*

_{M}*d*´

*d*]

^{(4)}´

*d*]

^{(2)}

*ñ .*

_{M}We
take a special interest in states with *J _{0}*
= 0, that is why we treat the state (

*J*= 2,

_{0}*J*= 2) mentioned above as equivalent to (

*J*= 4,

_{0}*J*= 2) and to (

*J*= 0,

_{0}*J*= 2) . Therefore we say, the configuration (

*J*¹ 0,

_{0}*J*= 2) does not exist. In a similar way we see that the

*J*= 3-states (

*J*= 2, 4) differ only in their signs. Both

_{0}*J*= 4-states (

*J*= 2, 4) are identical. For

_{0}*J*= 5 (

*J*= 4) the expression (3.6) vanishes.

_{0}*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 *n _{d}*

*d*-bosons is written as follows

| *n _{d}* , ([

*d*´

*d*]

^{(0)}

_{0})

^{n}*�—([[*

^{p}*d*´

*d*]

^{(2)}´

*d*]

^{(0)}

_{0})

^{n}*�—(*

^{D}*d*

*)*

^{l}^{(J)}

*ñ . (3.8)*

_{M}In the
expression (3.8) the doublet [*d* ´ *d*]^{(}^{0)}_{0
}with angular momentum 0 appears *n*_{p}_{ }times and the
triplet [[*d* ´ *d*]^{(2)} ´ *d*]^{(0)}_{0}
exists *n** _{D}*
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

*n*= 2

_{d}*n*

*p*+ 3

*n*

*+*

_{D}*l*. The number

*t*=

*n*- 2

_{d}*n*

*p*= 3

*n*

*+*

_{D}*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}^{)}

*the "reduced" state of the*

_{M}*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**

*n*,

_{d}*n*

*p*,

*n*

_{D},*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, ... , 2*l* - 3, 2*l* - 2, 2*l*, (3.9)

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

The
exclusion of *J* = 2*l* - 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* = 2*l* holds. We now construct the
symmetric state with *M*
= 2*l* - 1 and
represent it using numbered bosons whose projections of the angular momentum is
*m *:

|(*d*^{ }* ^{l}*)

_{M}_{ = 2}

_{l}_{ - 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* = 2*l* - 1. On the other hand, if one
turns the "stretched" state ( with *J* = 2*l *) relative
to the z-axis in order to obtain the projection *M* = 2*l* - 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* = 2*l* - 1 is not allowed because its
maximal projection would be *M* = 2*l* - 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

*J*= 0 which is inconsistent with the term "reduced" state. The rule (3.9) excludes

_{0}*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 *n _{d}* the allowed values

*n*

*and*

_{p}*n*

*are given. Accompanying values for*

_{D}*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. *n _{d}*
: number of

*d*-bosons,

*n*

*: number of boson pairs with total angular momentum 0,*

_{p}*n*

*: number of boson triplets with total angular momentum 0,*

_{D}*t*: seniority,

*l*: number of bosons in the "reduced" state

*,*

*J*: total angular momentum

n_{d} |
n_{p } |
t = n - 2_{d}n_{p } |
n_{D } |
l = t -3n_{D } |
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 *n _{d}*
> 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 ( *n _{d}*,

*t*,

*J*)-values differ in the quantity

*n*

*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*

_{D}*n*

*looses its character of an ordering number and it has to be replaced by an arbitrarily defined index.*

_{D}Many-boson
configurations in the seniority scheme stand out because they are eigenfunctions of the vibrational limit of the *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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