Introduction to the Graphical Theory of Angular Momentum: Case Studies

230 Pages·2009·1.86 MB·English

by  Ewald Balcar

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Ewald Balcar Stephen W. Lovesey
·
Introduction to the
Graphical Theory
of Angular Momentum
Case Studies
ABC
Prof.Dr.EwaldBalcar Prof.Dr.StephenW.Lovesey
ViennaUniversityofTechnology ISISFacility,STFC,&DiamondLight
AtominstitutderO¨sterreichischen SourceLtd
Universita¨ten DidcotOX110QXOxfordshire
Stadionallee2 England,UK
1020Vienna [email protected]
Austria
[email protected]
E.Balcar,S.W.Lovesey,IntroductiontotheGraphicalTheoryofAngularMomentum:Case
Studies,STMP234(Springer,BerlinHeidelberg2009),DOI10.1007/978-3-642-03118-2
ISSN0081-3869 e-ISSN1615-0430
ISBN978-3-642-03117-5 e-ISBN978-3-642-03118-2
DOI10.1007/978-3-642-03118-2
SpringerHeidelbergDordrechtLondonNewYork
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(cid:2)c Springer-VerlagBerlinHeidelberg2009
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Dedicated in gratitude to
Hildegard and Margaret.
Acknowledgements
It is our pleasure to express our gratitude to Prof. Dimitri A. Varshalovich
for his interest and communications in the initial stage of our work. In a way,
his impressive work provided the stimulus and touchstone for the subject of
this book.
It is also obvious that, without the generous continuing support and in-
frastructure of our home institutions through many years this book would
never have been compiled. Oneof us, E.B., wishes to expresshisappreciation
of encouragement and friendship extended by Prof. Helmut Rauch as direc-
tor of the Atominstitut Vienna, while S.W.L. is grateful to A.D. Taylor and
N. Gidopoulos (ISIS Facility) and G. Materlik (Diamond Light Source Ltd.)
for ongoing support.
Editorial assistance
WewanttoexpressourgratitudetothestaffofSpringerVerlagwho,inmany
ways, have helped to create this book, and to acknowledge, in this respect,
thesupportreceivedfromAdelheidDuhmandJaquelineLenz,whosuggested
technical adaptations of the final manuscript.
TeXnical assistance
In preparing this work we have benefited a great deal from the contributions
of many people, too many to name them all. The foundation was laid by
Donald E. Knuth who, driven by his quest for beauty in fonts, type-setting
and book-printing, has presented the world of science with TeX.
This book, in particular, would have been very difficult to write without
the PS-Tricks package of Timothy van Zandt which was used extensively to
create the diagrams. We have used the TeXLive distribution and LaTeX2e,
and the user interface was WinShell written and supported by Ingo de Boer.
Preface
Lectures on Quantum Mechanics will, in most cases, stop at Clebsch–Gordan
coefficients and, perhaps, mention the Wigner–Eckart Theorem. Any student
venturingbeyondthislimitintothetheoryofatomicstateswill,ofcourse,be
immediately confronted with further coupling coefficients of higher order.
Once familiar with these concepts and quantities, it is quite natural to
seek and explore the conditions and options available. Drawing from our own
experiences,werealizedthatatextlikethisintroductioncouldencourageand
facilitate getting deeper into the subject. However, an introductory chapter
presentsnotation,relationsbetweenandsymmetrypropertiesof,moreorless,
familiar coupling coefficients as a bridge-head into unknown territory.
The graphical representations and methods for treating and manipulating
higher-order coupling coefficients represent the main topic of this book. Al-
thoughthegraphicaltheorypresentedherehasbeenaroundforsometime,it
has, perhaps, not received the attention it merits.
Starting from the simplest building blocks we gradually introduce more
complex elements. Initially the emphasis is on the equivalence between al-
gebraic and graphical representations, but later we shift to solving problems
with the help of graphical methods. The solutions to these problems are de-
rived in a step-by-step manner to provide the option of self-study. Most of
the results are known from the literature, but it would be negligent not to
demonstratethatthepowerofthegraphicalmethodenablesonetotranscend
the set of known results, as we do in a few cases.
The final, and rather difficult, chapter is devoted to an application of the
graphical theory within a calculation relevant in the area of atomic physics.
Therewedemonstratetheeleganceandsuperiorityofthegraphicalmethodin
deriving a result for which the algebraic method is only a theoretical option.
Thisbookcontains,perhaps,lesswrittentextthanmightbeexpectedbut
manymorefigures.Itisthusobviousthatthediagramsrepresentanessential
part of the presentation and merit detailed study. We have tried hard to
provide enough explanatory figure captions to soften the impact of complex
diagrams.
X Preface
Only the future will tell whether we were successful in compiling a useful
and reader-friendly book, as was our intention. Although great care was ex-
ercised in the preparation of text and diagrams, we have to accept the blame
for all errors escaping unnoticed in the final copy.
Vienna and Wantage Ewald Balcar
June 2009 Stephen W. Lovesey
PS: Looking back, it could be said, this project may have started a long
time ago, when we had first contact with coupling coefficients in quantum
mechanics. For one of us (E.B.) it was his teacher, mentor and friend Prof.
Otto Hittmair, who introduced him to these strange entities, that govern
even stranger nuclear reactions. In later years both authors had the privilege
of working as postdocs with Walter Marshall (later Lord Marshall of Gor-
ing), and again, we encountered these vector coupling coefficients as essential
ingredientsofmany-electronwavefunctionsdescribingmagneticneutronscat-
tering.
On and off through the years, due to our mutual influence and collabora-
tion, familiarity with the vector coupling coefficients of higher rank increased
and culminated in the developments of a program1 for the algebraic values of
these coefficients as a tool for theoretical physicists working in this field.
It was only in recent past, when the focus of our work shifted from neu-
trons to x-rays, that we became aware of the potential and promise offered
by mastering the graphical technique for manipulating vector coupling coef-
ficients. In order to profit from this potential one has, however, to suffer the
labours of the plain acquiring enough skills before attempting to climb. The
urge to facilitate the hard work by providing a ‘how-to’ manual for mastering
the graphical theory of angular momentum supplied the primary momentum
for assembling the material collected in this book.
1 In order to obtain this Windows program please contact one of the authors by
e-mail.
Contents
1 Introduction............................................... 1
1.1 Historical Background ................................... 1
1.2 Intent and Motivation.................................... 2
1.3 Layout and Contents..................................... 3
1.4 Application of the Graphical Theory....................... 5
2 Properties of 3jm- and 3nj-Symbols....................... 7
2.1 Properties of 3jm-Symbols ............................... 7
2.2 Properties of 3nj-Symbols ................................ 10
2.3 Comment on Notation ................................... 10
3 Basic Tools for the Graphical Method ..................... 11
3.1 Graphical Representation of 3jm-Symbols .................. 11
3.2 Special Elements of the Graphical Method.................. 13
3.3 Treatment of 3jm-Symbols with a Zero-line................. 14
3.4 Direct Change of a Node Sign............................. 15
3.5 Single External Line ..................................... 17
3.6 Two-line Nodes ......................................... 19
3.7 A Simple Closed Diagram ................................ 21
3.8 Reversal of Direction for an Internal Line................... 24
3.9 Line Reversal of an External Line ......................... 25
3.9.1 Reversal of an Outgoing Line ....................... 26
3.9.2 Reversal of an Ingoing Line......................... 26
3.9.3 Reversal of an Outgoing Line and a j =0-Line........ 27
3.9.4 Reversal of an Ingoing Line and a j =0-Line ......... 28
4 Orthogonality Relations for 3jm-Symbols.................. 31
4.1 Identity for 3jm-Symbols I ............................... 31
4.2 Identity for 3jm-Symbols II............................... 34
4.3 Identity for 3jm-Symbols III.............................. 36
XII Contents
5 Properties of the 6j-Symbol ............................... 41
5.1 Definition of the 6j-Symbol............................... 41
5.2 Symmetry Properties of the 6j-Symbol..................... 43
5.3 6j-Symbol with a zero element ............................ 45
5.4 6j-Symbol with Three 0-Lines............................. 46
6 Properties of the 9j-Symbol ............................... 49
6.1 Definition of the 9j-Symbol............................... 49
6.2 Symmetry Properties of the 9j-Symbol..................... 51
6.3 A 9j-Symbol with a zero element.......................... 52
7 General Principles for Diagrams ........................... 55
7.1 Multiplication Convention ................................ 55
7.2 Notation for Graphical Elements .......................... 56
7.3 Transformation and Deformation .......................... 56
7.4 Diagrams Connected by a Single Internal Line .............. 57
7.5 Separation for Diagrams Connected by Two Lines ........... 58
7.6 Contraction of Two Angular Momentum Lines .............. 60
7.7 Summation Procedure ................................... 61
7.8 Separation for Three Lines ............................... 63
7.9 Joining Two Diagrams ................................... 66
7.10 Separation for Four Lines................................. 67
7.11 Separation for Five Lines................................. 69
8 Closed Diagrams .......................................... 73
8.1 Sums with Single 3nj-Symbols ............................ 73
8.1.1 Sum with a 3j-Symbol ............................. 73
8.1.2 Alternating Sum with a 3j-Symbol .................. 75
8.1.3 Sum with a 6j-Symbol ............................. 76
8.1.4 Alternating Sum with a 6j-Symbol .................. 78
8.1.5 Sum with a 9j-Symbol ............................. 81
8.2 Sums with Two 3nj-Symbols ............................. 82
8.2.1 Sum with Two 6j-Symbols ......................... 82
8.2.2 Alternating Sum with Two 6j-Symbols............... 85
8.2.3 Sum with a 6j- and a 9j-Symbol .................... 88
8.2.4 Sum with Two 9j-Symbols ......................... 93
8.3 Sums with Three 3nj-Symbols ............................ 99
8.3.1 The 9j-Symbol Expressed Through 6j-Symbols ....... 99
8.3.2 Sum with One 9j- and Two 6j-Symbols ..............103
8.3.3 Alternating Sum with One 9j- and Two 6j-Symbols ...109
9 Open Diagrams............................................119
9.1 Sum with Two 3jm-Symbols..............................119
9.2 Sum with Three 3jm-Symbols ............................122
9.3 Sum with Four 3jm-Symbols I ............................126