Group theoretical methods and applications to molecules and crystals:
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| Format: | Buch |
| Sprache: | English |
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[2005]
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| Ausgabe: | paperback ed. |
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| Beschreibung: | XVI, 492 S. graph. Darst. |
| ISBN: | 0521640628 9780521020381 |
Internformat
MARC
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| 100 | 1 | |a Kim, Shoon K. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Group theoretical methods and applications to molecules and crystals |c Shoon K. Kim |
| 250 | |a paperback ed. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c [2005] | |
| 300 | |a XVI, 492 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Kristallmathematik |0 (DE-588)4125615-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Gruppentheorie |0 (DE-588)4072157-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Kristallmathematik |0 (DE-588)4125615-3 |D s |
| 689 | 0 | 1 | |a Gruppentheorie |0 (DE-588)4072157-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016669625&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016669625 | ||
Datensatz im Suchindex
| _version_ | 1804137916269592576 |
|---|---|
| adam_text | Contents
Preface
page
xv
List of symbols
xvii
1
Linear transformations
1.1
Vectors
1
1.2
Linear transformations and matrices
2
1.2.1
Functions of a matrix
5
1.2.2
Special matrices
6
1.2.3
Direct products of matrices
6
1.2.4
Direct sums of matrices
7
1.3
Similarity transformations
8
1.3.1
Functions of a matrix (revisited)
8
1.4
The characteristic equation of a matrix
9
1.4.1
Diagonalizability and projection operators
10
1.5
Unitary transformations and normal matrices
12
1.5.1
Examples of normal matrices
13
1.6
Exercises
14
2
The theory of matrix transformations
2.1
Involutional transformations
17
2.2
Application to the Dirac theory of the electron
20
2.2.1
The Dirac y-matrices
20
2.2.2
The Dirac plane waves
21
2.2.3
The symmetric Dirac plane waves
23
2.3
Intertwining matrices
24
2.3.1
Idempotent matrices
26
2.4
Matrix diagonalizations
27
2.5
Basic properties of the characteristic transformation matrices
30
2.6
Construction of a transformation matrix
31
2.7
Illustrative examples
34
3
Elements of abstract group theory
3.1
Group axioms
37
3.1.1
The criterion for a finite group
38
3.1.2
Examples of groups
38
3.2
Group generators for a finite group
40
3.2.1
Examples
42
3.3
Subgroups and coset decompositions
43
3.3.1
The criterion for subgroups
44
vi
Contents
3.3.2 Lagrange s
theorem
44
3.4
Conjugation and classes
45
3.4.1
Normalizers
45
3.4.2
The centralizer
46
3.4.3
The center
47
3.4.4
Classes
47
3.5
Isomorphism and homomorphism
48
3.5.1
Examples
49
3.5.2
Factor groups
50
3.6
Direct products and semidirect products
51
4
Unitary and orthogonal groups
4.1
The unitary group U(n)
53
4.1.1
Basic properties
53
4.1.2
The exponential form
54
4.2
The orthogonal group O{n, c)
55
4.2.1
Basic properties
55
4.2.2
Improper rotation
56
4.2.3
The real orthogonal group O(n, r)
57
4.2.4
Real exponential form
58
4.3
The rotation group in three dimensions
0(3,
r)
58
4.3.1
Basic properties of rotation
58
4.3.2
The conjugate rotations
62
4.3.3
The
Euler
angles
63
5
The point groups of finite order
5.1
Introduction
66
5.1.1
The
uniaxial
group Cn
67
5.1.2 Multiaxial
groups. The equivalence set of axes and axis-vectors
67
5.1.3
Notations and the multiplication law for point operations
68
5.2
The dihedral group Dn
73
5.3
Proper polyhedral groups Po
75
5.3.1
Proper cubic groups,
Tand O
77
5.3.2
Presentations of polyhedral groups
79
5.3.3
Subgroups of proper point groups
82
5.3.4
Theorems on the axis-vectors of proper point groups
83
5.4
The Wyle theorem on proper point groups
85
5.5
Improper point groups
86
5.5.1
General discussion
86
5.5.2
Presentations of improper point groups
88
5.5.3
Subgroups of point groups of finite order
90
5.6
The angular distribution of the axis-vectors of rotation for regular
polyhedral groups
91
5.6.1
General discussion
91
5.6.2
Icosahedral group
Y
93
5.6.3
Buckminsterfullerene C6o (buckyball)
97
5.7
Coset enumeration
98
Contents
vii
6
Theory of group
representations
6.1
Hubert spaces and linear operators
102
6.1.1
Hubert spaces
102
6.1.2
Linear operators
103
6.1.3
The matrix representative of an operator
105
6.2
Matrix representations of a group
107
6.2.1
Homomorphism conditions
107
6.2.2
The regular representation
109
6.2.3
Irreducible representations
110
6.3
The basis of a group representation
112
6.3.1
The carrier space of a representation
112
6.3.2
The natural basis of a matrix group
114
6.4
Transformation of functions and operators
115
6.4.1
General discussion
115
6.4.2
The group of transformation operators
117
6.4.3
Transformation of operators under
G
=
{R} 1
19
6.5
Schur s lemma and the orthogonality theorems on irreducible
representations
120
6.6
The theory of characters
125
6.6.1
Orthogonality relations
125
6.6.2
Frequencies and irreducibility criteria
126
6.6.3
Group functions
127
6.7
Irreducible representations of point groups
128
6.7.1
The group Cn
129
6.7.2
The group Dn
129
6.7.3
The group
Γ
131
6.7.4
The group
О
133
6.7.5
The improper point groups
135
6.8
Properties of irreducible bases
135
6.8.1
The orthogonality of basis functions
135
6.8.2
Application to perturbation theory
136
6.9
Symmetry-adapted functions
138
6.9.1
Generating operators
138
6.9.2
The projection operators
141
6.10
Selection rules
144
7
Construction of symmetry-adapted linear combinations based on
the correspondence theorem
7.1
Introduction
149
7.2
The basic development
150
7.2.1
Equivalent point space S(n)
150
7.2.2
The correspondence theorem on basis functions
152
7.2.3
Mathematical properties of bases on S(n)
153
7.2.4
Illustrative examples of the SALCs of equivalent scalars
155
7.3
SALCs of equivalent
orbitais
in general
158
7.3.1
The general expression of SALCs
158
7.3.2
Two-point bases and operator bases
160
7.3.3
Notations for equivalent
orbitais
161
viii Contents
7.3.4
Alternative elementary bases
161
7.3.5
Illustrative examples
162
7.4
The general classification of
S
ALCs
166
7.4.1
U^Ã)
SALCs from the equivalent
orbitais
Є ІЇА)
X
A^
167
7.5
Hybrid atomic
orbitais
170
7.5.1
The
σ
-bonding
hybrid AOs
171
7.5.2
General hybrid AOs
173
7.6
Symmetry coordinates of molecular vibration based on the correspon¬
dence theorem
174
7.6.1
External symmetry coordinates of vibration
175
7.6.2
Internal vibrational coordinates
177
7.6.3
Illustrative examples
179
8
Subduced and induced representations
8.1
Subduced representations
188
8.2
Induced representations
189
8.2.1
Transitivity of induction
191
8.2.2
Characters of induced representations
191
8.2.3
The irreducibility condition for induced representations
191
8.3
Induced representations from the
irreps
of
anormal
subgroup
193
8.3.1
Conjugate representations
193
8.3.2
Little groups and orbits
194
8.3.3
Examples
195
8.4
Irreps
of a solvable group by induction
197
8.4.1
Solvable groups
197
8.4.2
Induced representations for a solvable group
198
8.4.3
Case I (reducible)
199
8.4.4
Case II (irreducible)
201
8.4.5
Examples
202
8.5
General theorems on induced and subduced representations and
construction of unirreps via small representations
203
8.5.1
Induction and
subduction
203
8.5.2
Small representations of a little group
205
8.5.3
Induced representations from small representations
206
9
Elements of continuous groups
9.1
Introduction
209
9.1.1
Mixed continuous groups
210
9.2
The Hurwitz integral
211
9.2.1
Orthogonality relations
215
9.3
Group generators and Lie algebra
215
9.4
The connectedness of a continuous group and the multivalued
representations
219
10
The representations of the rotation group
10.1
The structure of SU(2)
224
10.1.1
The generators of
Щ2)
224
10.1.2
The parameter space Q ofSU(2)
226
Contents ix
10.1.3 Spinors 228
10.1.4
Quaternions
228
10.2
The homomorphism between
Щ2)
and SO(3, r)
229
10.3
Unirreps W e) of the rotation group
232
10.3.1
The homogeneity of
ľP](S)
233
10.3.2
Theumtarityofű^S)
234
10.3.3
TheirreducibilityofD^ÍÖ)
234
10.3.4
The completeness of the unirreps
{£^(0);
j = OĄ,l,...}
235
10.3.5
Orthogonality relations of
1УЈ)(в)
235
10.3.6
The
Hurwitz
density function for St/(2)
236
10.4
The generalized spinors and the angular momentum eigenfunctions
237
10.4.1
The generalized spinors
237
10.4.2
The transformation of the total angular momentum eigenfunc¬
tions under the general rotation Uj
238
10.4.3
The vector addition model
240
10.4.4
The Ciebsch-Gordan coefficients
241
10.4.5
The angular momentum eigenfunctions for one electron
245
11
Single- and double-valued representations of point groups
11.1
The double-valued representations of point groups expressed by the
projective
representations
247
11.1.1
The
projective
set of a point group
247
11.1.2
The orthogonality relations for
projective
unirreps
248
11.2
The structures of double point groups
250
11.2.1
Defining relations of double point groups
250
11.2.2
The structure of the double dihedral group
D „
251
11.2.3
The structure of the double octahedral group O
253
11.3
The unirreps of double point groups expressed by the
projective
unirreps
of point groups
257
11.3.1
The
uniaxial
group
Oc
257
11.3.2
The group Cn
258
11.3.3
The group A«
258
11.3.4
The group
£)„ 260
11.3.5
The group
О
261
11.3.6
The tetrahedral group
T
263
12
Projective
representations
12.1
Basic concepts
266
12.2
Projective
equivalence
268
12.2.1
Standard factor systems
269
12.2.2
Normalized factor systems
270
12.2.3
Groups of factor systems and multiplicators
271
12.2.4
Examples of
projective
representations
272
12.3
The orthogonality theorem on
projective
irreps
274
12.4
Covering groups and representation groups
276
12.4.1
Covering groups
276
12.4.2
Representation groups
278
χ
Contents
12.5
Representation groups of double point groups
279
12.5.1
Representation groups of double proper point groups P
279
12.5.2
Representation groups of double rotation-inversion
groups P i
281
12.6
Projective
unirreps of double rotation-inversion point groups P
283
12.6.1
The
projective
unirreps of
Сѓгј
285
12.6.2
The
projective
unirreps of
D n¡
286
12.6.3
The
projective
unirreps of O
287
13
The
230
space groups
13.1
The Euclidean group in three dimensions EPi
289
13.2
Introduction to space groups
293
13.3
The general structure of
Bravais
lattices
295
13.3.1
Primitive bases
295
13.3.2
The projection operators for
a Bravais
lattice
298
13.3.3
Algebraic expressions for the
Bravais
lattices
299
13.4
The
14
Bravais
lattice types
302
13.4.1
The hexagonal system
H
(D6¡)
302
13.4.2
The tetragonal system
Q
(D4i)
303
13.4.3
The rhombohedral system
RH
(Ą,)
306
13.4.4
The orthorhombic system
O (Ą,)
307
13.4.5
The cubic system
С (О;)
308
13.4.6
The monoclinic system
M
(Сг;)
309
13.4.7
The triclinic system
T (Q)
311
13.4.8
Remarks
311
13.5
The
32
crystal classes and the lattice types
313
13.6
The
32
minimal general generator sets for the
230
space groups
315
13.6.1
Introduction
315
13.6.2
The space groups of the class £>4
316
13.7
Equivalence criteria for space groups
318
13.8
Notations and defining relations
321
13.8.1
Notations
321
13.8.2
Defining relations of the crystal classes
322
13.9
The space groups of the cubic system
323
13.9.1
The class
T
326
13.9.2
The class
T¡
(=Гћ)
327
13.9.3
The class
О
329
13.9.4
The class Tp (=Td)
330
13.9.5
The class
Q
(=( )
331
13.10
The space groups of the rhombohedral system
332
13.10.1
The class C3
333
13.10.2
The class C3i
333
13.10.3
The class
Ą
333
13.10.4
The class C3v
334
13.10.5
The class
Ą,
(=DM)
335
13.11
The hierarchy of space groups in a crystal system
336
13.11.1
The cubic system
337
13.11.2
The hexagonal system
337
Contents xi
13.11.3
The rhombohedral system
337
13.11.4
The tetragonal system
338
13.12
Concluding remarks
338
14
Representations of the space groups
14.1
The unirreps of translation groups
340
14.2
The reciprocal lattices
342
14.2.1
General discussion
342
14.2.2
Reciprocal lattices of the cubic system
344
14.2.3
The Miller indices
345
14.2.4
The density of lattice points on a plane
346
14.3
Brillouin zones
347
14.3.1
General construction of Brillouin zones
347
14.3.2
The wave vector point groups
348
14.3.3
The Brillouin zones of the cubic system
349
14.4
The small representations of wave vector space groups
352
14.4.1
The wave vector space groups Gk
352
14.4.2
Small representations of Gk via the
projective
representations
ofGi
354
14.4.3
Examples of the small representations of Gk
356
14.5
The unirreps of the space groups
358
14.5.1
The irreducible star
359
14.5.2
A summary of the induced representation of the space
groups
360
15
Applications of unirreps of space groups to energy bands and
vibrational modes of crystals
15.1
Energy bands and the eigenmnctions of an electron in a crystal
362
15.2
Energy bands and the
eigeníunctions
for the free-electron model in a
crystal
365
15.2.1
The notations for a small representation of Gk
368
15.2.2
Example
1.
A simple cubic lattice
368
15.2.3
Example
2.
The diamond crystal
371
15.3
Symmetry coordinates of vibration of a crystal
377
15.3.1
General discussion
3 77
15.3.2
The small representations of the wave vector groups Gk based
on the equivalent Bloch functions
379
15.4
The symmetry coordinates of vibration for the diamond crystal
382
15.4.1
General discussion
382
15.4.2
Construction of the symmetry coordinates of vibration
385
16
Time reversal, anti-unitary point groups and their
co-representations
16.1
Time-reversal symmetry, classical
391
16.1.1
General introduction
391
16.1.2
The time correlation function
393
1
6.1.3
Onsager s reciprocity relation for transport coefficients
394
xii
Contents
16.2
Time-reversal symmetry, quantum mechanical
396
16.2.1
General introduction
396
16.2.2
The properties of the time-reversal operator
0 400
16.2.3
The time-reversal symmetry of matrix elements of a physical
quantity
401
16.3
Anti-unitary point groups
403
16.3.1
General discussion
403
16.3.2
The classification of ferromagnetics and ferroelectrics
407
16.4
The co-representations of anti-unitary point groups
409
16.4.1
General discussion
409
16.4.2
Three types of co-unirreps
410
16.5
Construction of the co-unirreps of anti-unitary point groups
413
16.5.1
G
414
16.5.2
Q, C andC*
415
16.5.3
DenanáDl
417
16.5.4
The cubic groups
418
16.6
Complex conjugate representations
419
16.7
The orthogonality theorem on the co-unirreps
422
16.8
Orthogonality relations for the characters, the irreducibility condition
and the type criteria for co-unirreps
424
16.8.1
Orthogonality relations for the characters of co-unirreps
424
16.8.2
Irreducibility criteria for co-unirreps
424
16.8.3
The type criterion for a co-unirrep
425
17
Anti-unitary space groups and their co-representations
17.1
Introduction
428
17.2
Anti-unitary space groups of the first kind
430
17.2.1
The cubic system
432
17.2.2
The hexagonal system
433
17.2.3
The rhombohedral system
433
17.2.4
The tetragonal system
433
17.2.5
The orthorhombic system
433
17.2.6
The monoclinic system
435
17.2.7
The triclinic system
435
17.3
Anti-unitary space groups of the second kind
438
17.3.1
Illustrative examples
441
17.3.2
Concluding remarks
441
17.4
The type criteria for the co-unirreps of anti-unitary space groups and
anti-unitary wave vector groups
442
17.5
The representation groups of anti-unitary point groups
445
17.6
The
projective co-unirreps
of anti-unitary point groups
450
17.6.1
Examples for the construction of the
projective
co-unirreps of H
458
17.7
The co-unirreps of anti-unitary wave vector space groups
460
17.7.1
Concluding remarks
463
17.8
Selection rules under an anti-unitary group
464
17.8.1
General discussion
464
Contents xiii
17.8.2 Transitions
between states belonging to different co-unirreps
465
17.8.3
Transitions between states belonging to the same co-unirrep
467
17.8.4
Selection rules under a gray point group
470
Appendix. Character tables for the crystal point groups
472
References
483
Index
487
|
| adam_txt |
Contents
Preface
page
xv
List of symbols
xvii
1
Linear transformations
1.1
Vectors
1
1.2
Linear transformations and matrices
2
1.2.1
Functions of a matrix
5
1.2.2
Special matrices
6
1.2.3
Direct products of matrices
6
1.2.4
Direct sums of matrices
7
1.3
Similarity transformations
8
1.3.1
Functions of a matrix (revisited)
8
1.4
The characteristic equation of a matrix
9
1.4.1
Diagonalizability and projection operators
10
1.5
Unitary transformations and normal matrices
12
1.5.1
Examples of normal matrices
13
1.6
Exercises
14
2
The theory of matrix transformations
2.1
Involutional transformations
17
2.2
Application to the Dirac theory of the electron
20
2.2.1
The Dirac y-matrices
20
2.2.2
The Dirac plane waves
21
2.2.3
The symmetric Dirac plane waves
23
2.3
Intertwining matrices
24
2.3.1
Idempotent matrices
26
2.4
Matrix diagonalizations
27
2.5
Basic properties of the characteristic transformation matrices
30
2.6
Construction of a transformation matrix
31
2.7
Illustrative examples
34
3
Elements of abstract group theory
3.1
Group axioms
37
3.1.1
The criterion for a finite group
38
3.1.2
Examples of groups
38
3.2
Group generators for a finite group
40
3.2.1
Examples
42
3.3
Subgroups and coset decompositions
43
3.3.1
The criterion for subgroups
44
vi
Contents
3.3.2 Lagrange's
theorem
44
3.4
Conjugation and classes
45
3.4.1
Normalizers
45
3.4.2
The centralizer
46
3.4.3
The center
47
3.4.4
Classes
47
3.5
Isomorphism and homomorphism
48
3.5.1
Examples
49
3.5.2
Factor groups
50
3.6
Direct products and semidirect products
51
4
Unitary and orthogonal groups
4.1
The unitary group U(n)
53
4.1.1
Basic properties
53
4.1.2
The exponential form
54
4.2
The orthogonal group O{n, c)
55
4.2.1
Basic properties
55
4.2.2
Improper rotation
56
4.2.3
The real orthogonal group O(n, r)
57
4.2.4
Real exponential form
58
4.3
The rotation group in three dimensions
0(3,
r)
58
4.3.1
Basic properties of rotation
58
4.3.2
The conjugate rotations
62
4.3.3
The
Euler
angles
63
5
The point groups of finite order
5.1
Introduction
66
5.1.1
The
uniaxial
group Cn
67
5.1.2 Multiaxial
groups. The equivalence set of axes and axis-vectors
67
5.1.3
Notations and the multiplication law for point operations
68
5.2
The dihedral group Dn
73
5.3
Proper polyhedral groups Po
75
5.3.1
Proper cubic groups,
Tand O
77
5.3.2
Presentations of polyhedral groups
79
5.3.3
Subgroups of proper point groups
82
5.3.4
Theorems on the axis-vectors of proper point groups
83
5.4
The Wyle theorem on proper point groups
85
5.5
Improper point groups
86
5.5.1
General discussion
86
5.5.2
Presentations of improper point groups
88
5.5.3
Subgroups of point groups of finite order
90
5.6
The angular distribution of the axis-vectors of rotation for regular
polyhedral groups
91
5.6.1
General discussion
91
5.6.2
Icosahedral group
Y
93
5.6.3
Buckminsterfullerene C6o (buckyball)
97
5.7
Coset enumeration
98
Contents
vii
6
Theory of group
representations
6.1
Hubert spaces and linear operators
102
6.1.1
Hubert spaces
102
6.1.2
Linear operators
103
6.1.3
The matrix representative of an operator
105
6.2
Matrix representations of a group
107
6.2.1
Homomorphism conditions
107
6.2.2
The regular representation
109
6.2.3
Irreducible representations
110
6.3
The basis of a group representation
112
6.3.1
The carrier space of a representation
112
6.3.2
The natural basis of a matrix group
114
6.4
Transformation of functions and operators
115
6.4.1
General discussion
115
6.4.2
The group of transformation operators
117
6.4.3
Transformation of operators under
G
=
{R} 1
19
6.5
Schur's lemma and the orthogonality theorems on irreducible
representations
120
6.6
The theory of characters
125
6.6.1
Orthogonality relations
125
6.6.2
Frequencies and irreducibility criteria
126
6.6.3
Group functions
127
6.7
Irreducible representations of point groups
128
6.7.1
The group Cn
129
6.7.2
The group Dn
129
6.7.3
The group
Γ
131
6.7.4
The group
О
133
6.7.5
The improper point groups
135
6.8
Properties of irreducible bases
135
6.8.1
The orthogonality of basis functions
135
6.8.2
Application to perturbation theory
136
6.9
Symmetry-adapted functions
138
6.9.1
Generating operators
138
6.9.2
The projection operators
141
6.10
Selection rules
144
7
Construction of symmetry-adapted linear combinations based on
the correspondence theorem
7.1
Introduction
149
7.2
The basic development
150
7.2.1
Equivalent point space S(n)
150
7.2.2
The correspondence theorem on basis functions
152
7.2.3
Mathematical properties of bases on S(n)
153
7.2.4
Illustrative examples of the SALCs of equivalent scalars
155
7.3
SALCs of equivalent
orbitais
in general
158
7.3.1
The general expression of SALCs
158
7.3.2
Two-point bases and operator bases
160
7.3.3
Notations for equivalent
orbitais
161
viii Contents
7.3.4
Alternative elementary bases
161
7.3.5
Illustrative examples
162
7.4
The general classification of
S
ALCs
166
7.4.1
U^Ã)
SALCs from the equivalent
orbitais
Є ІЇА)
X
A^
167
7.5
Hybrid atomic
orbitais
170
7.5.1
The
σ
-bonding
hybrid AOs
171
7.5.2
General hybrid AOs
173
7.6
Symmetry coordinates of molecular vibration based on the correspon¬
dence theorem
174
7.6.1
External symmetry coordinates of vibration
175
7.6.2
Internal vibrational coordinates
177
7.6.3
Illustrative examples
179
8
Subduced and induced representations
8.1
Subduced representations
188
8.2
Induced representations
189
8.2.1
Transitivity of induction
191
8.2.2
Characters of induced representations
191
8.2.3
The irreducibility condition for induced representations
191
8.3
Induced representations from the
irreps
of
anormal
subgroup
193
8.3.1
Conjugate representations
193
8.3.2
Little groups and orbits
194
8.3.3
Examples
195
8.4
Irreps
of a solvable group by induction
197
8.4.1
Solvable groups
197
8.4.2
Induced representations for a solvable group
198
8.4.3
Case I (reducible)
199
8.4.4
Case II (irreducible)
201
8.4.5
Examples
202
8.5
General theorems on induced and subduced representations and
construction of unirreps via small representations
203
8.5.1
Induction and
subduction
203
8.5.2
Small representations of a little group
205
8.5.3
Induced representations from small representations
206
9
Elements of continuous groups
9.1
Introduction
209
9.1.1
Mixed continuous groups
210
9.2
The Hurwitz integral
211
9.2.1
Orthogonality relations
215
9.3
Group generators and Lie algebra
215
9.4
The connectedness of a continuous group and the multivalued
representations
219
10
The representations of the rotation group
10.1
The structure of SU(2)
224
10.1.1
The generators of
Щ2)
224
10.1.2
The parameter space Q'ofSU(2)
226
Contents ix
10.1.3 Spinors 228
10.1.4
Quaternions
228
10.2
The homomorphism between
Щ2)
and SO(3, r)
229
10.3
Unirreps W\e) of the rotation group
232
10.3.1
The homogeneity of
ľP](S)
233
10.3.2
Theumtarityofű^S)
234
10.3.3
TheirreducibilityofD^ÍÖ)
234
10.3.4
The completeness of the unirreps
{£^(0);
j = OĄ,l,.}
235
10.3.5
Orthogonality relations of
1УЈ)(в)
235
10.3.6
The
Hurwitz
density function for St/(2)
236
10.4
The generalized spinors and the angular momentum eigenfunctions
237
10.4.1
The generalized spinors
237
10.4.2
The transformation of the total angular momentum eigenfunc¬
tions under the general rotation Uj
238
10.4.3
The vector addition model
240
10.4.4
The Ciebsch-Gordan coefficients
241
10.4.5
The angular momentum eigenfunctions for one electron
245
11
Single- and double-valued representations of point groups
11.1
The double-valued representations of point groups expressed by the
projective
representations
247
11.1.1
The
projective
set of a point group
247
11.1.2
The orthogonality relations for
projective
unirreps
248
11.2
The structures of double point groups
250
11.2.1
Defining relations of double point groups
250
11.2.2
The structure of the double dihedral group
D'„
251
11.2.3
The structure of the double octahedral group O'
253
11.3
The unirreps of double point groups expressed by the
projective
unirreps
of point groups
257
11.3.1
The
uniaxial
group
Oc
257
11.3.2
The group Cn
258
11.3.3
The group A«
258
11.3.4
The group
£)„ 260
11.3.5
The group
О
261
11.3.6
The tetrahedral group
T
263
12
Projective
representations
12.1
Basic concepts
266
12.2
Projective
equivalence
268
12.2.1
Standard factor systems
269
12.2.2
Normalized factor systems
270
12.2.3
Groups of factor systems and multiplicators
271
12.2.4
Examples of
projective
representations
272
12.3
The orthogonality theorem on
projective
irreps
274
12.4
Covering groups and representation groups
276
12.4.1
Covering groups
276
12.4.2
Representation groups
278
χ
Contents
12.5
Representation groups of double point groups
279
12.5.1
Representation groups of double proper point groups P'
279
12.5.2
Representation groups of double rotation-inversion
groups P'i
281
12.6
Projective
unirreps of double rotation-inversion point groups P\
283
12.6.1
The
projective
unirreps of
Сѓгј
285
12.6.2
The
projective
unirreps of
D'n¡
286
12.6.3
The
projective
unirreps of O\
287
13
The
230
space groups
13.1
The Euclidean group in three dimensions EPi
289
13.2
Introduction to space groups
293
13.3
The general structure of
Bravais
lattices
295
13.3.1
Primitive bases
295
13.3.2
The projection operators for
a Bravais
lattice
298
13.3.3
Algebraic expressions for the
Bravais
lattices
299
13.4
The
14
Bravais
lattice types
302
13.4.1
The hexagonal system
H
(D6¡)
302
13.4.2
The tetragonal system
Q
(D4i)
303
13.4.3
The rhombohedral system
RH
(Ą,)
306
13.4.4
The orthorhombic system
O (Ą,)
307
13.4.5
The cubic system
С (О;)
308
13.4.6
The monoclinic system
M
(Сг;)
309
13.4.7
The triclinic system
T (Q)
311
13.4.8
Remarks
311
13.5
The
32
crystal classes and the lattice types
313
13.6
The
32
minimal general generator sets for the
230
space groups
315
13.6.1
Introduction
315
13.6.2
The space groups of the class £>4
316
13.7
Equivalence criteria for space groups
318
13.8
Notations and defining relations
321
13.8.1
Notations
321
13.8.2
Defining relations of the crystal classes
322
13.9
The space groups of the cubic system
323
13.9.1
The class
T
326
13.9.2
The class
T¡
(=Гћ)
327
13.9.3
The class
О
329
13.9.4
The class Tp (=Td)
330
13.9.5
The class
Q
(=(\)
331
13.10
The space groups of the rhombohedral system
332
13.10.1
The class C3
333
13.10.2
The class C3i
333
13.10.3
The class
Ą
333
13.10.4
The class C3v
334
13.10.5
The class
Ą,
(=DM)
335
13.11
The hierarchy of space groups in a crystal system
336
13.11.1
The cubic system
337
13.11.2
The hexagonal system
337
Contents xi
13.11.3
The rhombohedral system
337
13.11.4
The tetragonal system
338
13.12
Concluding remarks
338
14
Representations of the space groups
14.1
The unirreps of translation groups
340
14.2
The reciprocal lattices
342
14.2.1
General discussion
342
14.2.2
Reciprocal lattices of the cubic system
344
14.2.3
The Miller indices
345
14.2.4
The density of lattice points on a plane
346
14.3
Brillouin zones
347
14.3.1
General construction of Brillouin zones
347
14.3.2
The wave vector point groups
348
14.3.3
The Brillouin zones of the cubic system
349
14.4
The small representations of wave vector space groups
352
14.4.1
The wave vector space groups Gk
352
14.4.2
Small representations of Gk via the
projective
representations
ofGi
354
14.4.3
Examples of the small representations of Gk
356
14.5
The unirreps of the space groups
358
14.5.1
The irreducible star
359
14.5.2
A summary of the induced representation of the space
groups
360
15
Applications of unirreps of space groups to energy bands and
vibrational modes of crystals
15.1
Energy bands and the eigenmnctions of an electron in a crystal
362
15.2
Energy bands and the
eigeníunctions
for the free-electron model in a
crystal
365
15.2.1
The notations for a small representation of Gk
368
15.2.2
Example
1.
A simple cubic lattice
368
15.2.3
Example
2.
The diamond crystal
371
15.3
Symmetry coordinates of vibration of a crystal
377
15.3.1
General discussion
3 77
15.3.2
The small representations of the wave vector groups Gk based
on the equivalent Bloch functions
379
15.4
The symmetry coordinates of vibration for the diamond crystal
382
15.4.1
General discussion
382
15.4.2
Construction of the symmetry coordinates of vibration
385
16
Time reversal, anti-unitary point groups and their
co-representations
16.1
Time-reversal symmetry, classical
391
16.1.1
General introduction
391
16.1.2
The time correlation function
393
1
6.1.3
Onsager's reciprocity relation for transport coefficients
394
xii
Contents
16.2
Time-reversal symmetry, quantum mechanical
396
16.2.1
General introduction
396
16.2.2
The properties of the time-reversal operator
0 400
16.2.3
The time-reversal symmetry of matrix elements of a physical
quantity
401
16.3
Anti-unitary point groups
403
16.3.1
General discussion
403
16.3.2
The classification of ferromagnetics and ferroelectrics
407
16.4
The co-representations of anti-unitary point groups
409
16.4.1
General discussion
409
16.4.2
Three types of co-unirreps
410
16.5
Construction of the co-unirreps of anti-unitary point groups
413
16.5.1
G\
414
16.5.2
Q, C'andC*
415
16.5.3
DenanáDl
417
16.5.4
The cubic groups
418
16.6
Complex conjugate representations
419
16.7
The orthogonality theorem on the co-unirreps
422
16.8
Orthogonality relations for the characters, the irreducibility condition
and the type criteria for co-unirreps
424
16.8.1
Orthogonality relations for the characters of co-unirreps
424
16.8.2
Irreducibility criteria for co-unirreps
424
16.8.3
The type criterion for a co-unirrep
425
17
Anti-unitary space groups and their co-representations
17.1
Introduction
428
17.2
Anti-unitary space groups of the first kind
430
17.2.1
The cubic system
432
17.2.2
The hexagonal system
433
17.2.3
The rhombohedral system
433
17.2.4
The tetragonal system
433
17.2.5
The orthorhombic system
433
17.2.6
The monoclinic system
435
17.2.7
The triclinic system
435
17.3
Anti-unitary space groups of the second kind
438
17.3.1
Illustrative examples
441
17.3.2
Concluding remarks
441
17.4
The type criteria for the co-unirreps of anti-unitary space groups and
anti-unitary wave vector groups
442
17.5
The representation groups of anti-unitary point groups
445
17.6
The
projective co-unirreps
of anti-unitary point groups
450
17.6.1
Examples for the construction of the
projective
co-unirreps of H'
458
17.7
The co-unirreps of anti-unitary wave vector space groups
460
17.7.1
Concluding remarks
463
17.8
Selection rules under an anti-unitary group
464
17.8.1
General discussion
464
Contents xiii
17.8.2 Transitions
between states belonging to different co-unirreps
465
17.8.3
Transitions between states belonging to the same co-unirrep
467
17.8.4
Selection rules under a gray point group
470
Appendix. Character tables for the crystal point groups
472
References
483
Index
487 |
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| any_adam_object_boolean | 1 |
| author | Kim, Shoon K. |
| author_facet | Kim, Shoon K. |
| author_role | aut |
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| illustrated | Illustrated |
| index_date | 2024-07-02T21:40:10Z |
| indexdate | 2024-07-09T21:19:53Z |
| institution | BVB |
| isbn | 0521640628 9780521020381 |
| language | English |
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| oclc_num | 903504263 |
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| physical | XVI, 492 S. graph. Darst. |
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| spelling | Kim, Shoon K. Verfasser aut Group theoretical methods and applications to molecules and crystals Shoon K. Kim paperback ed. Cambridge [u.a.] Cambridge Univ. Press [2005] XVI, 492 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Kristallmathematik (DE-588)4125615-3 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Kristallmathematik (DE-588)4125615-3 s Gruppentheorie (DE-588)4072157-7 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016669625&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kim, Shoon K. Group theoretical methods and applications to molecules and crystals Kristallmathematik (DE-588)4125615-3 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| subject_GND | (DE-588)4125615-3 (DE-588)4072157-7 |
| title | Group theoretical methods and applications to molecules and crystals |
| title_auth | Group theoretical methods and applications to molecules and crystals |
| title_exact_search | Group theoretical methods and applications to molecules and crystals |
| title_exact_search_txtP | Group theoretical methods and applications to molecules and crystals |
| title_full | Group theoretical methods and applications to molecules and crystals Shoon K. Kim |
| title_fullStr | Group theoretical methods and applications to molecules and crystals Shoon K. Kim |
| title_full_unstemmed | Group theoretical methods and applications to molecules and crystals Shoon K. Kim |
| title_short | Group theoretical methods and applications to molecules and crystals |
| title_sort | group theoretical methods and applications to molecules and crystals |
| topic | Kristallmathematik (DE-588)4125615-3 gnd Gruppentheorie (DE-588)4072157-7 gnd |
| topic_facet | Kristallmathematik Gruppentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016669625&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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