123doc

Symmetry

Symmetry refers to the repetition of shapes and dimensions in objects, observable from various perspectives, and is a pervasive element in our lives In art and architecture, symmetry enhances aesthetic appeal, while in classical music, composers like J.S Bach utilize symmetrical structures by repeating themes with variations Additionally, symmetry plays a crucial role in science, influencing the fundamental physical and chemical properties of objects.

Symmetry is a fundamental concept in science, influencing theories, models, and the shapes of discrete objects Nature inherently favors symmetry, which is essential for comprehending everything from elementary particles to galaxy structures Most natural laws stem from various forms of symmetry, allowing us to predict the physical properties of systems by analyzing their symmetrical aspects This principle is particularly vital in understanding the physical and chemical properties of solids.

A transformation occurs when a change is made to a physical system, whether it's a tangible object or a mathematical representation of a physical property If the system remains unchanged before and after this transformation, it is termed invariant The symmetry of a system encompasses all transformation operations that maintain this invariance, which is crucial for both classical and quantum physics Understanding these symmetries aids in comprehending the properties of matter at both atomic and macroscopic scales Importantly, the laws of physics governing a system must retain their invariance under these symmetry transformations; for instance, the Hamiltonian operator, which represents the total energy in quantum mechanics, must remain invariant under any symmetry operation applied to the system.

A spatial symmetry transformation acts about asymmetry element A symmetry element can be a point, an axis, or a plane of symmetry resulting in inversion, rotation,

Navajo weavers are renowned for their ability to create intricate patterns that exhibit symmetry along both horizontal and vertical axes, all without relying on pre-drawn designs or mirror transformations For instance, a two-dimensional array featuring four identical atoms positioned at the corners of a square, as illustrated in Fig 1.2, highlights this concept In this arrangement, the center point of the square serves as a symmetry element for an inversion operation.

The arrangement of atoms in a square exhibits symmetry through rotational and reflective operations A central axis perpendicular to the plane of the square serves as a symmetry element, allowing for 90, 180, 270, and 360-degree rotations that maintain the atom configuration For instance, a 90-degree counterclockwise rotation shifts atoms from points 1 to 2, 2 to 3, 3 to 4, and 4 to 1 Additionally, there are four mirror planes that intersect the symmetry axis, with two bisecting the square's sides and the other two passing through opposite corners A specific mirror plane from point 1 to point 3 keeps those atoms unchanged while swapping the positions of atoms at points 2 and 4 In three-dimensional space, combined symmetry elements can occur, involving rotation about an axis followed by reflection in a perpendicular plane.

Mathematical operations on physical systems can be represented using matrices, while the physical properties of matter are described by tensors of varying ranks An nth rank tensor in three-dimensional space consists of n indices and 3^n components, adhering to specific transformation rules A zero-rank tensor, known as a scalar, has no indices; a first-rank tensor, or vector, has one index and three components; and a second-rank tensor, referred to as a matrix, has two indices and nine components Higher-order tensors extend this classification Group theory and tensor algebra are essential mathematical frameworks that elucidate the symmetry properties of physical systems Although group theory is a valuable tool in physics, enabling the identification of various physical properties without extensive calculations, it primarily offers qualitative insights rather than precise predictions regarding the magnitude of these properties.

Fig 1.2 Symmetry elements 4 of a square array of four equivalent atoms

Crystal Structures

Solids are categorized into two types: amorphous and crystalline Glass exemplifies an amorphous structure, lacking long-range order, yet it possesses short-range symmetry at the molecular level In contrast, crystalline solids exhibit long-range order characterized by translational symmetry.

Crystals are structured as three-dimensional, periodic arrangements of atoms or molecules, characterized by a distinct lattice and basis The repeating group of atoms or molecules is known as a basis or unit cell, with the smallest unit referred to as a primitive unit cell Primitive translation vectors define the dimensions of a primitive unit cell, while the array of points formed by these vectors is termed a lattice.

The equation T n ẳn1aỵn2bỵn3c; (1.1) represents a lattice structure where a, b, and c are the primitive translation vectors, and the ni are integers This arrangement ensures that the configuration of atoms or molecules appears identical from any lattice point.

In a crystal structure, each atom in the basis corresponds to a specific lattice point, although not all atoms are positioned directly on these points For instance, in a two-dimensional square lattice, atoms A occupy each lattice point, while atoms B are situated between the lattice points at the positions marked as T*B = (1/2)(a + b) This arrangement illustrates the relationship between atoms and lattice points in a simple lattice model.

Any operation performed on a crystal that carries the crystal structure into itself is part of the symmetry group for that crystal This may include translations,

The fundamental types of crystal lattices are characterized by their symmetry operations, which include translation group, point group, and space group symmetries The translation group consists of operations represented by fEjT*, where E denotes the identity rotation operation and T* signifies a translation operation that preserves the crystal's structure Examples of these translation operations include the primitive lattice vectors T* = 1a*, T* = 1b*, and T* = 1a* + 2b* In contrast, the point group encompasses operations defined by {a|0}, where 'a' represents a symmetry operation at a point that maintains the crystal's invariance without any translation A crystal's lattice point group can exhibit two-, three-, four-, or sixfold axes of rotation, along with reflections and inversion operations.

In the example illustrated in Fig 1.3, the symmetry operations at lattice point A are derived from those in Fig 1.2 The space group consists of operations that maintain the crystal's invariance A space group is classified as symmorphic if all operations form a subgroup The point group operations from the array of equivalent atoms in Fig 1.2, when combined with the lattice translation operations in Fig 1.3, create a symmorphic space group However, if the atoms are not all equivalent, as seen in Fig 1.4, the space group may not be symmorphic, necessitating the combination of translations with point group operations For instance, in Fig 1.4a, the 90°, 180°, and 270° rotations remain symmetry operations, but the four mirror planes require translation to maintain invariance, leading to the definition of glide planes Similarly, in Fig 1.4b, the 90° and 270° rotations only preserve the system's invariance when paired with specific translation operations, resulting in screw axes Notably, glide plane and screw axis symmetry operations in three dimensions have restrictions not found in the two-dimensional cases discussed.

In crystallography, the combined operation of symmetry elements, such as glide and screw axes, requires specific alignment with the mirror plane or rotation axis For instance, a three-dimensional atomic arrangement can be represented in the s13 plane, as shown in Fig 1.4a In Fig 1.4b, a 180-degree rotation about the 1–3 axis maintains the array's invariance only when paired with a translation along the 1–3 axis, indicating the presence of a screw axis Nature features 14 distinct types of crystal lattices, known as Bravais lattices, characterized by their primitive translation vectors and the angles between them—specifically, angles a, b, and g The lattice parameters denote the lengths of these vectors, and each lattice type is depicted in Fig 1.5, categorized into seven unique crystal systems, each with its own geometric shape The relationships between the lattice parameters and angles are detailed in Table 1.1, with the Bravais unit cell representing the smallest symmetrical unit cell of the structure.

The primitive Bravais lattices consist of a single lattice site located at (0,0,0), while there are three types of nonprimitive Bravais lattices The two-face-centered lattice, labeled as C, includes lattice sites at (0,0,0) and (1/2,1/2,0) The internally centered lattice, designated as I, has lattice sites at (0,0,0) and (1/2,1/2,1/2) The all-face-centered lattice, known as F, features lattice sites at (0,0,0), (1/2,1/2,0), (1/2,0,1/2), and (0,1/2,1/2) In total, there are seven types of primitive Bravais lattices and seven types of nonprimitive lattices, which collectively form the fourteen crystal symmetries Additionally, when examining space groups, it is essential to categorize the two-face-centered lattice structure into three distinct types.

Table 1.1 Three-dimensional crystal lattices

Crystal systems and Bravais lattices

Point group symmetry (Crystal classes)

Fig 1.5 Crystal structures depending on which faces have the centered lattice points These are designated

A (1/2bþ1/2c), B (1/2cþ1/2a), and C (1/2aþ1/2b) where the coordinates of the face center are given in parenthesis.

Each crystal system is defined by a unique point group symmetry, which distinguishes it from other systems, as outlined in Chapter 2 This symmetry, presented in Table 1.1, represents the maximum symmetry elements of the simple Bravais lattice for each crystal class Subgroups listed in parentheses indicate lower symmetry point groups, arising when the molecules or atoms on the Bravais lattice exhibit less symmetry than the lattice itself For an in-depth exploration of these point groups, refer to Chapter 2.

The triclinic lattice has two types of point groups, while the monoclinic and orthorhombic systems each feature three types The tetragonal and hexagonal systems consist of seven types each, and both the trigonal and cubic systems have five types In total, there are 32 point groups corresponding to the 14 Bravais lattices, as detailed in Table 1.1.

Crystals exhibit five types of symmetry axes corresponding to n-fold rotations: one-fold (C1), two-fold (C2), three-fold (C3), four-fold (C4), and six-fold (C6), each representing a rotation of 360/n degrees Additionally, there are mirror planes: those perpendicular to the major rotation axis (h), those containing the major rotation axis (v), and diagonal mirror planes (sd) Identity and inversion operations are denoted as E and i, while combined rotation/reflection operations with a perpendicular mirror plane are labeled Sn Point groups are categorized using Schoenflies notation, where groups with n-th order cyclic rotations are represented as Cn Adding a mirror reflection normal to the rotation axis results in Cnh groups, while Cnv groups have mirror planes that include the rotation axis The introduction of a two-fold rotation element perpendicular to the major rotation axis leads to Dn groups, and Dnd groups possess additional vertical symmetry planes Additionally, tetrahedral groups (T, Td, Th) and octahedral groups (O, Oh) are recognized, with their corresponding lattice systems illustrated in Figure 1.5.

The triclinic system is unique in that it lacks both rotation and reflection symmetry elements, making it the least symmetric of all lattice systems It can be described as a parallelepiped with unequal edges and angles.

The monoclinic system is characterized by a twofold axis in the b-direction and a mirror plane that is perpendicular to this axis In this system, the unit cell features three edges of unequal lengths, with two edges being perpendicular to the symmetry axis This structure can manifest as either a simple primitive lattice or a base-centered cell, which includes points located at the center of the faces that are parallel to the reflection plane.

Three mutually orthogonal twofold axes identify an orthorhombic system.

The unit cell features reflection planes that are perpendicular to its axes, with three edges that are unequal in length yet mutually orthogonal, resulting in four types of Bravais lattices: primitive, base-centered, body-centered, and face-centered The tetragonal system is characterized by a fourfold rotation axis, alongside twofold axes and mirror planes typical of the orthorhombic system, due to the equality of two edges in the lattice cell, leading to both primitive and body-centered lattices In contrast, the trigonal system is defined by a threefold rotation axis, where the lattice cell takes the shape of a rhombohedron with equal sides and angles, none of which are 90 degrees This system also includes three twofold rotation axes perpendicular to the trigonal axis, an inversion operation, three reflection planes containing the trigonal axis, and two combined rotations of 60 degrees about the trigonal axis with reflection in a perpendicular plane Additionally, there exists a second type of lattice in the trigonal system, designated as P, which is equivalent to the hexagonal lattice.

The hexagonal system is defined by its characteristic sixfold symmetry axis, complemented by threefold and twofold rotational axes Additionally, it features twofold rotation axes that are perpendicular to the main axis and exhibits inversion symmetry In the lattice cell, two edges have equal lengths, while two angles measure 90 degrees, with the third angle differing.

120 This has only the primitive lattice structure.

Symmetry in Reciprocal Space

Quasiparticles on a periodic crystal lattice (such as electrons or phonons) are described by eigenfunctions of the form

Table 1.2 List of the 230 space groups

Crystal system Schoenflies notation International notation

To analyze wave vectors with lattice periodicity, it is essential to construct a reciprocal lattice This involves defining three vectors, b1, b2, and b3, based on the primitive translation vectors The relationships are given by b1 = t2t3 / (t1 · (t2 × t3)), b2 = t3t1 / (t1 · (t2 × t3)), and b3 = t1t2 / (t1 · (t2 × t3)) Here, k represents the wave vector with a magnitude of 2π/l, where l is the wavelength, and u(r) is a periodic function corresponding to the lattice structure.

The volume of the unit cell in reciprocal space is represented by b1 ã (b2 b3), which is the inverse of the volume in ordinary space, t1ã (t2 t3) This framework allows for the formation of unit cells in reciprocal space corresponding to each Bravais lattice in ordinary space, known as Brillouin zones To construct these zones, vectors K that define the reciprocal lattice are drawn, and each vector is bisected with planes perpendicular to K The shape formed by these bisecting planes is identified as the first Brillouin zone This zone is then replicated throughout the reciprocal lattice by translating it using reciprocal lattice vectors, ensuring that any point k within a specific Brillouin zone is equivalent to the corresponding point defined by k in the first Brillouin zone.

A wave vector that has the correct periodicity in reciprocal space can then be expressed as

In reciprocal space, the vector \( K \) is expressed as \( K = h_1b_1 + h_2b_2 + h_3b_3 \), where \( h \) represents integers This formulation indicates that the vectors extend from the origin to lattice points The relationship \( e^{iK \cdot (r - R_n)} = e^{iK \cdot r} \) holds true, highlighting the significance of the reciprocal space vector Furthermore, the values of \( k \) within the first Brillouin zone are constrained by the condition \( p a < k < p a \), where \( a \) denotes the unit cell dimension in real space, and the endpoints of \( k \) lie on the zone surface The interaction between the reciprocal space vector and a unit cell vector in real space is also crucial for understanding the underlying physical properties.

* aiẳ2phi; (1.8) wherehiis an integer The function in (1.2) with the periodicity of the lattice can then be written as uð * rị ẳX

Since the Bravais lattice is invariant with respect to the symmetry elements of its point group, the corresponding reciprocal lattice must also be invariant with respect

Table 1.3 Equivalent lattices in real and reciprocal space

Lattice in real space Lattice in reciprocal space

One-face-centered monoclinic One-face-centered monoclinic

Face-centered orthorhombic Body-centered orthorhombic

Body-centered orthorhombic Face-centered orthorhombic

One-face-centered orthorhombic One-face-centered orthorhombic

Body-centered tetragonal Body-centered tetragonal

Face-centered cubic Body-centered cubic

Body-centered cubic Face-centered cubic

The Brillouin zone for a simple cubic lattice with space group O 1 h is illustrated in Fig 1.10, highlighting points of special symmetry associated with this point group While the Brillouin zone corresponds to the same crystal system as the real-space lattice, its distribution pattern may differ For instance, a face-centered cubic lattice in real space translates to a body-centered cubic lattice in reciprocal space For further details, refer to Table 1.3, which lists the equivalent lattices in both real and reciprocal spaces.

The Brillouin zone, illustrated in Fig 1.10, represents the reciprocal lattice cell for the simple cubic space group O 1 h in ordinary space This figure highlights special symmetry points, each associated with specific point group symmetries: Oh for G and R, D4h for M and X, C4v for D and T, C3v for L, and C2v for Z, S, and S.

Figure 1.11 illustrates a second example of a Brillouin zone, specifically for a hexagonal crystal system The diagram highlights the points of special symmetry within the Brillouin zone Comprehensive structures for Brillouin zones across all crystal systems can be found in reference [3].

Translational symmetry and reciprocal space are especially important in considering wave-like quasiparticles in crystals Examples of this for phonons and electrons are given in Chaps.7and8, respectively. k z k 2 k 1

Fig 1.11 Brillouin zone for a hexagonal crystal structure

Problems

1 There are 10 point groups in two dimensions with the following symmetry operations:

Draw two-dimensional figures representing each of these symmetry groups.

Create stereograms for each of the ten two-dimensional point groups mentioned Illustrate how elongating the C4v shape from problem 1 alters its symmetry elements in the stereogram, resulting in a new stereogram that represents a different point group.

3 A two-dimensional hexagonal lattice is shown in the picture below.

To illustrate the primitive unit cell for the given lattice, one must first draw the Wigner–Seitz cell, which represents the region where all points are closer to a designated lattice point than to any other This symmetric cell effectively partitions the lattice into distinct areas, allowing for the analysis of its geometric properties To complete the task, calculate the areas of both the primitive unit cell and the Wigner–Seitz cell, providing insight into their respective spatial characteristics.

4 Calculate the volume of the conventional unit cell and the primitive unit cell for a simple cubic lattice, a body-centered cubic lattice, and a face-centered cubic lattice.

5 Look up the crystal structure of each of the following materials: NaCl, CsCl, ZnS, Al 2 O 3 , and Diamond.

1 C Kittel, Introduction to Solid State Physics (Wiley, New York, 1957)

2 W.A Harrison, Solid State Theory (McGraw-Hill, New York, 1970)

3 M Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, New York, 1974)

4 B DiBartolo, R.C Powell, Phonons and Resonances in Solids (Wiley, New York, 1976)

5 International Union of Crystallography, International Tables for X-Ray Crystallography (Kynoch, Birmingham, 1952)

6 M Sachs, Solid State Theory (McGraw-Hill, New York, 1963)

7 J.C Slater, Symmetry and Energy Bands in Crystals (Dover, New York, 1972)

Group theory is the formal mathematical framework for understanding the symmetry of physical systems, as introduced in Chapter 1 This chapter outlines the essential properties of group theory that will be applied to various physical examples in later sections The focus of this book is on the practical applications of group theory, rather than the derivation of fundamental principles or advanced topics For a comprehensive exploration of group theory, readers are encouraged to consult the referenced material.

A group is a collection of elements that adhere to specific criteria and interact according to a defined rule, commonly referred to as the "multiplication" of the elements This interaction may differ from standard multiplication, as the elements involved are not limited to simple numbers The total number of elements in a group is known as the order of the group For a set of elements to qualify as a group, it must meet four essential requirements.

1 One element, designatedEand called the identity element, commutes with all the other elements of the group and multiplication of an element byEleaves the element unchanged That is,EAẳAEẳA.

2 The result of multiplying any two elements in a group (including the product of an element with itself) is an element of the group That is,ABẳCwhereA,B, and

Care all elements of the group.

3 Every element of the group must have a reciprocal element that is also an element of the group That is, ARẳRAẳEwhereAis an element of the group,

Ris its reciprocal, andEthe identity element andRandEare both members of the group.

4 The associative law of multiplication is valid for the product of any three elements of the group That is,A(BC)ẳ(AB)C.

In a group, the product of its elements does not always adhere to the commutative law, meaning that the result of multiplying two elements, AB, may differ from the result of BA However, if the elements in a particular group do follow the commutative law, the group is classified as Abelian.

The properties of a group can be illustrated using a group multiplication table, specifically one that includes six elements labeled A, B, C, D, E, and F In this table, each entry represents the product of the element from its column and the element from its row, confirming that the identity element is part of the group, the product of any two elements remains within the group, and every element has an inverse within the group Notably, each element appears only once in each row and column While the associative law is satisfied, the commutative law does not apply universally, indicating that the group is non-Abelian, with an order of 6.

The multiplication table is instrumental in identifying subgroups within a larger group, which are subsets that fulfill the criteria of a group independently of the other elements For instance, the elements D, E, and F constitute a subgroup of order 3, while there are three distinct subgroups of order 2: E, A; E, B; and E, C Additionally, the element E alone forms a subgroup of order 1 It's important to note that the orders of these subgroups are integral factors of the total group's order.

A valuable approach for managing a group is to arrange its components into conjugate pairs using similarity transformations To determine the conjugate of an element A, one can compute the triple product of A with another group element and its reciprocal.

In group theory, elements A and B are considered conjugates if they undergo a similarity transformation, with every element being conjugate to itself Furthermore, if A is conjugate to both B and C, then B and C are also conjugate to each other, forming a class within the group Analyzing the multiplication table of elements A, B, C, D, E, and F reveals that element E constitutes a class of order 1, while elements A, B, and C together form a class of order 3, as demonstrated by the various similarity transformations applied to element A.

E 1 AEẳA; A 1 AAẳA; B 1 ABẳC; C 1 ACẳB; D 1 ADẳB;

The elements F, B, C, D, and F can undergo various similarity transformations, demonstrating that elements D and F belong to a class of order 2 It is important to note that the order of any class within a group is always an integral factor of the group's overall order.

The focus of this article is on asymmetry groups, which consist of a complete set of symmetry operations that adhere to group rules Specifically, the article highlights the symmetry groups that define the crystal classes mentioned in Chapter 1.

Basic Concepts of Group Theory

Group theory's fundamental concepts can be illustrated through the spatial symmetry of geometrically shaped objects This symmetry, known as point group symmetry, describes how an object transforms when subjected to various operations around a specific point in space The key symmetry operations associated with point groups include rotations about axes, reflections across planes, inversions through a central point, and their combinations.

Different types of symmetry elements are designated by specific notations The identity operation, where no transformation occurs, is labeled as E Rotation around an axis of symmetry is represented by Cn, indicating that the object appears identical after a rotation of 2π/n For instance, C2 signifies a 180-degree rotation, while C4 denotes a 90-degree rotation Since n rotations of Cn return the object to its original position, Cn^n equals E A reflection plane perpendicular to the highest order symmetry axis is designated as sh, while sv indicates a reflection plane containing the highest order symmetry axis Diagonal mirror planes to the rotation axes are referred to as sd Mirror operations, which result in E after two applications, are essential in symmetry analysis For objects with a center of symmetry, the inversion operation is noted as i, where i^2 equals E Combined operations, such as inversion represented as i = C2sh, and improper rotations, denoted by Sn, which are a mix of rotation and reflection, illustrate the complexity of symmetry operations The sequence of these operations is crucial, as not all symmetry operations commute.

Organizing the elements of a group into classes is beneficial, as it allows for the classification of elements that are interconnected through a unitary transformation of another operator within the group For instance, if T1AT = A0, then both A and A0 belong to the same class, highlighting their relationship within the group.

As stated before, the order of a class must be an integral factor of the order of the group.

The action of the elements of a symmetry group on the physical properties of a system is described in terms of mathematical transformations The physical proper-

In quantum mechanics, state vectors form vector spaces, and their transformations mirror the symmetry transformations of the system's coordinates When the mathematical representation of a system's physical properties aligns with a symmetry group, it is termed a representation of that group Symmetry elements function as linear operators, facilitating transformations within a specific group representation Each group possesses various representations linked to distinct physical properties, including a one-dimensional trivial representation that assigns the value one to all group elements Typically, matrices of a specific dimension are associated with group elements to create a representation, adhering to the same multiplication rules as the group The representation's matrix is square, with its row or column count reflecting the representation's dimension and degeneracy, allowing for the construction of multiple representations for the same group.

It is always possible to find a similarity transformation that puts a matrix into a box diagonal form

In this caseAandA 0 are matrices representingreducible representationswhile the

Ai are matrices represent irreducible representations The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order of the group:

The number of irreducible representations of a group is equal to the number of classes in the group.

The spatial position of an object is defined by vectors in Cartesian coordinates, which can be transformed to represent the object's movement from an initial position (x, y, z) to a new position (x', y', z') Each vector can be expressed using unit vectors ^x, ^y, and ^z, allowing for transformation operations to be applied to individual components After these transformations, the new components are recombined to form the transformed vector r' Specifically, if a rotation occurs around the major symmetry axis, typically the z-axis, the transformation equations are x' = x cos(y) + y sin(y), y' = x sin(y) + y cos(y), and z' = z.

In matrix form this coordinate transformation is written as x 0 y 0 z 0

Aẳ ax 0 x ax 0 y ax 0 z ay 0 x ay 0 y ay 0 z az 0 x az 0 y az 0 z

The matrix elements ai 0 j represent the direction cosines of the coordinate system i 0 in relation to the coordinate system j, as illustrated in Fig 2.1 In the case of a rotation around the z-axis, the transformation matrix can be derived from equation (2.3).

* ðC yðzị ị ẳ cosy siny 0 siny cosy 0

A symmetry operation consisting of a mirror reflection plane perpendicular to thez- axis would be represented by the matrix

A; (2.6) z’ z y’ y x x’ cos –1 a x’z cos –1 a y’z cos –1 a z’z Fig 2.1 Transformation of

Cartesian coordinates so that the transformation is x 0 ẳx y 0 ẳy z 0 ẳ z:

Every element of a symmetry group can be represented by a transformation matrix such as the examples given in (2.5) and (2.6).

In matrix mathematics, the trace of a matrix is a significant property defined as the sum of its diagonal elements This concept is illustrated through examples of transformation matrices, highlighting its importance in various applications.

* ðC yðzị ị ẳX i aiiẳ2 cosyỵ1 (2.8) and

The trace of a transformation matrix associated with a symmetry operation is referred to as the character of the operation in that representation, denoted by w Characters of matrix operators possess unique properties that are particularly beneficial for applications in group theory.

1 Since the trace of a matrix is invariant under a similarity transformation, all symmetry operations belonging to the same class of the group have the same character.

2 The character of a reducible representation is equal to the sum of the characters of the irreducible representations that it contains.

3 The number of times that a specific irreducible representation is contained in the reduction of a reducible representation can be determined by n ðiị ẳ1 h

The character of an operation in an irreducible representation, denoted as Herew ðiị A, contrasts with the character of the same operation in a reducible representation, represented as w A This comparison involves summing over all symmetry operations within the group's order.

4 For any irreducible representation, the sum of the squares of the characters of all the operations equals the order of the group

5 The set of characters for two different irreducible representations are orthogonal

The direct product of two representations is determined by multiplying the characters associated with a specific operation in each representation, resulting in the character of that operation in the product representation Typically, a direct product representation is a reducible representation of the group.

Another important property of transformation matrices is that irreducible representations are orthogonal and obey the relationship

A ẵG i ðAị mn G j ðAị m 0 n 0 ẳ hffiffiffiffiffiffiffiffi didj p d ij d mm 0 d nn 0 : (2.13)

HereG i (A) mn is themnmatrix element of the transformation matrix for operation

Symmetry groups operate on sets of basis functions that transform into each other, representing specific physical properties of a system For instance, in coordinate transformations, the vector coordinates x, y, and z serve as basis functions Properties described by vectors transform according to the symmetry group's representation Additionally, the rotation axes Rx, Ry, and Rz can act as basis functions for irreducible representations, differing from spatial coordinates due to potential changes in rotation direction Another common set of basis functions includes the six components of a pseudovector derived from a vector product Spherical harmonic functions are also explored as basis functions in subsequent examples and discussions.

Character Tables

A character table for a symmetry group outlines the characters corresponding to each class of operations within the group and its irreducible representations The character tables for the 32 crystallographic point groups, as discussed in Chapter 1, are presented in Tables 2.1–2.32 These tables play a crucial role in applying group theory to analyze and determine the properties of crystals.

Table 2.3 Character table for point group C i

Table 2.4 Character table for point group for C 2

Table 2.5 Character table for point group C 2h

Table 2.6 Character table for point group C 2v

Table 2.7 Character table for point group D 2

Table 2.1 Character table for point group C 1

Table 2.2 Character table for point group C s

Table 2.8 Character table for point group D 2h

D 2h E C 2 ( z ) C 2 ( y ) C 2 ( x ) i s( xy ) s( xz ) s( yz ) Basis components

Table 2.12 Character table for point group S 4

Table 2.15 Character table for point group D 2d

D 2d E 2 S 4 C 2 2 C 0 2 2s d R 2 RS 4 RC 2 2 RC 0 2 2 Rs d Basis components

Table 2.21 Character table for point group S 6

Table 2.20 Character table for point group D 3d

Table 2.28 Character table for point group T

Table 2.29 Character table for point group T h

Table 2.31 Character table for point group O

Table 2.30 Character table for point group T d

Table 2.32 Character table for point group O h

In group theory, various notations are employed to denote irreducible representations, with G representing a generic representation The Mulliken notation is utilized in character tables to differentiate types of irreducible representations, where one-dimensional representations are indicated by A or B, depending on the character of the major rotation operation Two-dimensional representations are labeled E, while three-dimensional ones are designated T Subscripts 1 and 2 signify symmetric or antisymmetric twofold rotations relative to the principal rotation axis, and primes indicate symmetry concerning a horizontal plane For groups with inversion symmetry, subscripts g and u denote symmetric and antisymmetric representations, respectively Each character table features the point group identified by Schoenflies notation in the top left corner, followed by symmetry elements organized into classes and possible basis functions for irreducible representations The first column lists irreducible representations by increasing dimensions, while the main body presents the characters associated with symmetry elements for each representation, and the last column details the components of a vector or rotation basis function corresponding to that irreducible representation.

In various character tables, the two-dimensional E representation is denoted as E*, indicating that its characters are imaginary or complex To accurately interpret this representation, it is essential to decompose it into two distinct components.

Table 2.32 Character table for point group O h (continued)

O h R 8 RC 3 6 RC 2 6 RC 4 3 RC 2 4 Ri 6 RS 4 8 RS 6 3 Rs h 6 Rs d

In group theory, the number of irreducible representations of a group corresponds to the number of classes of elements within that group, as seen in the representation D 3/2u 4 1 0 0 0 4 0 1 0 0, where characters are complex conjugates To apply group theory to physical problems, it is essential that the characters are real; therefore, the sum of the characters from two complex representations is utilized to derive the characters of a real representation Specifically, for a rotation axis of order n, the complex character is expressed as e^(2πi/n) = cos(2πm/n) + i sin(2πm/n).

In the context of the point group C3, the expression e0 = e^n, e^n = n^2 = 1, and e^n = n^4 = i indicates that the double-valued representation E consists of two complex representations These representations have character sets for the classes E, C3, and C2, with values of 1, and ð1/2 + i√3.

=2ị Adding these gives the set of characters for the classes of theE* irreducible representation 2,1,1. Only the characters of the real representations are listed in the character tables.

In systems characterized by functions with half-integer values, it is essential to utilize double groups, leading to an increase in group order and a corresponding rise in the number of irreducible representations This scenario frequently arises in atomic physics when addressing spin or half-integer angular momentum For instance, the spin of an electron is represented by a function with two possible orientations relative to a quantization axis, and the Pauli spin operators that describe this phenomenon are represented by 2x2 matrices.

These are related to the angular momentum operatorJby sẳ2J:

Following the treatment ofJin quantum mechanics, the angular momentum rais- ing and lowering operators for spin can be expressed in terms of the Pauli spin operators [3].

The Pauli spin operators obey a multiplication table that has the properties of a group A rotation about an axisnin the two dimensional spin representation is given by the operator

Rð’;~nị ẳe ið1 = 2ị’sn ẳcos1

The operatorR(j,n) is also a 22 matrix.

An important result of (2.16) is that a rotation of 2pis not the identity operator for the group:

Rð’ỵ2p;nị ẳcosðpỵ’=2ị isnsinðpỵ’=2ị ẳ cosð’=2ị ỵisnsinð’=2ị ẳ Rð’;nị:

Instead an operator representing a rotation of 4pmust be introduced as the identity

Ewhile a rotation of 2pis a new operatorR ThenRmultiplied by all of the other operators of the group gives the additional group operators This leads to additional irreducible representations.

In group theory, spin is expressed through a two-dimensional irreducible representation known as G 1/2 Certain spatial operations exhibit differing characters for Cn and RCn, which are termed double valued The overall spatial and spin state of a system is depicted by the product of G 1/2 and the irreducible representations that define the system's spatial state Occasionally, this direct product yields additional new irreducible representations of the group.

D2d and Oh groups illustrate the additional elements and irreducible representations linked to double groups Chapter 4 delves deeper into these double-valued representations, providing examples for determining the characters of half-integer representations Understanding these concepts is crucial for analyzing magnetic properties and the impact of time reversal in quantum mechanical systems.

The irreducible representations for space groups are discussed in Chap.8.

Group Theory Examples

C 3v Point Group

To illustrate the application of group theory, let's examine an equilateral triangle, which exhibits six symmetry elements: the identity (E), a 120-degree rotation around the z-axis (C3), a 240-degree rotation (C2), and three mirror reflections (s1, s2, s3) through various planes This indicates that the group's order is 6 A multiplication table demonstrates that the product of any two elements remains within the group, each element has a reciprocal, and the associative law applies Consequently, all criteria for a mathematical group are satisfied.

The multiplication rules shown in Table 2.33 can be used to apply similarity transformations to these elements which allow them to be grouped into classes:

The elements C3 and C2 form one class with two symmetry elements, while the three mirror planes s1, s2, and s3 constitute another class Additionally, the element E stands alone as its own class Consequently, this group contains three classes, leading to three irreducible representations, with the sum of the squares of their dimensions equating to the group's order of 6 This configuration allows for two one-dimensional irreducible representations and one two-dimensional irreducible representation The character table for these representations can be developed in two ways: by expressing it in terms of unknown characters and utilizing the orthogonality of irreducible representations to derive the characters.

Fig 2.2 Equilateral triangle The z -axis direction is out of the page

Table 2.33 Multiplication table for equilateral triangle symmetry elements

The character of the identity operation is consistently equal to the dimension of the representation, indicating that there is always one totally symmetric irreducible representation where the character of each class is 1 Utilizing equation (2.11) yields the corresponding results.

X r g A 2 1 g A 2 2 ẳ ð1ỵ2aỵ3bị=6ẳ0∴2aỵ3bẳ 1 so aẳ1; bẳ 1;

X r gð ịgðEị ẳ ð2A1 ỵ2cỵ3dị=6ẳ0∴2cỵ3dẳ 2;

Combining the last two expressions givescẳ1 anddẳ0, so the character table is

The second way to derive the character table for this group is to consider how the Cartesian coordinates transform under the elements of the group In this case x 0 y 0 z 0

The article discusses a reducible representation G, which is detailed in the accompanying table The final entry in the table illustrates how the representation G can be expressed in terms of irreducible representations, utilizing the formula provided in equation (2.10).

For A1 this gives n ðA 1ị ẳ ð1=6ịð3ỵ0ỵ3ị ẳ1 For A2 it gives n (A2) ẳ(1/6) (3ỵ0–3)ẳ0 ForEit givesn (E) ẳ(1/6)(6ỵ0ỵ0)ẳ1.

The three transformation matrices found above have a box diagonal form

The upper left-hand corner boxes represent the matrices for the irreducible representation E, while the lower right-hand corner boxes correspond to the irreducible representation A1 The traces of these diagonal matrices provide the characters for both E and A1, with the characters for the A2 representation derived from the orthogonality condition Additionally, the transformation matrices for the symmetry elements reveal that the z component serves as the basis for the A1 irreducible representation, whereas the x and y components transform into combinations of one another according to the irreducible representation E, establishing (x, y) as the basis for E.

The rotation axis \( R_z \) remains unchanged under the operations of the \( E \) and \( C_3 \) classes but changes sign under the \( S \) operation, indicating it transforms according to the \( A_2 \) irreducible representation In contrast, the other rotation axes \( R_x \) and \( R_y \) transform into combinations of one another, forming a basis for the \( E \) irreducible representation Additionally, the transformation of vector components, specifically those of an axial vector formed by the product of two vectors, follows a conventional notation The transformation behavior of these components under the group's symmetry operations allows for the construction of transformation matrices for each symmetry element, with their traces calculated to ascertain the characters of the reducible representation.

The six-dimensional column matrix for a vector product is

Using this as a basis vector, the transformation vectors for an element of each class are written as

The irreducible representation can be expressed in terms of the 2E and 2A1 irreducible representations Notably, the basis function for one A1 representation is z², while (x² + y²) serves as the basis function for the other For the E representations, one set of basis functions is (xz, yz), and the other consists of (x²y², xy).

The character table for the symmetry group of an equilateral triangle, known as the point group C3v, encompasses all relevant information regarding basis functions.

In certain applications, it is crucial to compute the direct product of representations and subsequently reduce the results to the irreducible representations of the group For instance, when examining the group, the direct product of the E representation with itself involves squaring the character of the E representation for each symmetry class, yielding characters of 4, 1, and 0 for the E, C3, and s symmetry operations, respectively These characters correspond to a reducible representation, which can be analyzed using equation (2.10) to demonstrate that it simplifies to one E, one A1, and one A2 irreducible representation.

O h Point Group

In solid-state physics, a key symmetry is represented by a regular octahedron exhibiting center of inversion symmetry, which involves seven atoms positioned along the x, y, and z axes of a cube Each edge of the cube measures 2a in length, as illustrated in Fig 2.3, with the specific locations of the ions detailed in Table 2.34 The angle 'θ' is defined in the xy-plane, measured counterclockwise around the z-axis from the x-axis, while the angle 'φ' is determined in the xz-plane, measured counterclockwise around the y-axis from the z-axis.

Table 2.34 Ion positions in Fig 2.3 x y z r y ’

The cubicOhpoint group features 48 symmetry elements categorized into 10 classes, as detailed in the character table provided in Table 2.32 Key symmetry elements include the identity operation (E) and the inversion operation (i), each forming its own class Additionally, there are six C4 elements representing 90-degree rotations about the x, y, or z axes, along with three C2 axes of rotation for the same axes The group also includes six C2 0 axes running from the center of an edge to the center of the opposite edge, and eight axes for 120-degree rotations about the cube's body diagonals Furthermore, three mirror planes of symmetry intersect the centers of the cube's edges in the xy, xz, and yz planes, equivalent to combined C2i operations Six diagonal planes of symmetry correspond to combined C2 0i operations, while reflection operations can be combined with C3 and C4 rotations to yield eight S6 and six S4 operations, respectively.

The Oh point group consists of 10 irreducible representations, which correspond to its 10 distinct classes For the dimensions of these representations to sum up to the group's order of 48, the only feasible combination is 4(3)² + 2.

The group exhibits a total of four three-dimensional irreducible representations, two two-dimensional irreducible representations, and four one-dimensional representations, categorized into two sets of five One set corresponds to even parity under inversion, denoted by subscript g, while the other corresponds to odd parity, denoted by subscript u These representations can effectively operate on even and odd parity basis functions Notably, the character for symmetry operations that do not involve inversion remains consistent across both parity types However, for symmetry operations that include inversion, the character in the odd parity representation is equal to the character in the even parity representation multiplied by -1.

In the context of half-integer functions like spin, it is essential to introduce the additional operation R for a 2p rotation, as E represents a 4p rotation This adjustment leads to the emergence of three new g and three new u irreducible representations, detailed in Table 4.32.

In scenarios with high symmetry levels, such as the Oh group, utilizing subgroups can be advantageous A subgroup is defined as a subset of elements from the larger group that satisfies all the criteria to be considered a group itself For instance, D3d is a subgroup of Oh, which includes the identity element, two threefold rotation operations, three C2 0 operations, and the inversion operation applied to each of these elements.

D3d is given in Table2.30 The irreducible representations of the group can be

The decomposition of the T2g and A1g representations into combinations of irreducible representations of the subgroup reveals a correlation between the characters of common elements in the Oh and D3d groups This analysis highlights the relationship between the irreducible representations of the larger group and its subgroup, particularly for the even parity representations.

The inspection results can be validated by comparing them with the predictions outlined in equation (2.10) Specifically, the T2g irreducible representation of Oh will correspond to the A1g irreducible representation of D3d, appearing once, as indicated by the notation n(T2g) = 1.

12ð113ỵ210ỵ311ỵ113ỵ210ỵ311ị ẳ1; while theA2girreducible representation will appear the following number of times: n ðT 2g ị ẳ 1

This is consistent with the correlation table shown above.

Table 2.32 illustrates that the vector components (x, y, z) transform according to the T1 irreducible representations, which is crucial for applying group theory to identify permissible electromagnetic transitions, as elaborated in Section 2.4 and Chapter 4 Additionally, the table presents the irreducible representations for Oh that involve half-integer quantities, with Section 4.4 providing an example of their application for atoms possessing half-integer angular momentum values.

Group Theory in Quantum Mechanics

In quantum mechanics, a physical system is characterized by its Hamiltonian operator, H The system's allowed states are represented by a collection of orthonormal eigenfunctions, cn, while the corresponding energy levels are determined by a set of eigenvalues, En To obtain these eigenfunctions and eigenvalues, one must solve the Schrödinger equation.

The Hamiltonian, which characterizes a physical system, must remain unchanged under symmetry operations that preserve the system's invariance This principle is represented by a similarity transformation on the Hamiltonian, facilitated by a symmetry operator.

A symmetry operator that commutes with the Hamiltonian operator indicates that it leaves the Hamiltonian invariant, forming a group of symmetry operators Within this group, the identity operator represents an element that does not alter the Hamiltonian Additionally, for any two elements A and B in this group, their properties can be analyzed in relation to the symmetry of the system.

The relationship between AHA 1 ẳH and BHB 1 ẳH demonstrates that the product of two elements results in an invariant element Additionally, the associative law is upheld in this context Furthermore, when equation (2.19) is multiplied on the left by A and on the right by A 1, it yields significant results.

The inverse of an element in the context of the Hamiltonian leaves the system invariant, indicating that all elements maintaining this invariance adhere to group properties This collection of elements is known as the group associated with the Schrödinger equation, also referred to as the Hamiltonian group, and represents the symmetry group of the system defined by the Hamiltonian.

If one of the operators of the group of the HamiltonianAis applied to the initial Schro¨dinger equation given above,

The eigenfunctions transformed by the operator of the group H remain associated with the same eigenvalue En as the original eigenfunctions This indicates that jAcni is also an eigenfunction corresponding to the eigenvalue En, demonstrating the consistency of eigenfunction properties under transformation.

In quantum mechanics, the relationship between symmetry operators is highlighted by the equation A 1 = A†, which indicates that the inverse of a symmetry operator is equivalent to its adjoint This principle is essential for understanding the behavior of these operators within the framework of quantum theory, as demonstrated in equations (2.20) and (2.21).

An eigenvalue En is considered nondegenerate if it has only one eigenfunction, aside from a possible phase factor Conversely, when an eigenvalue is associated with an orthonormal set of eigenfunctions, it is termed degenerate In such cases, any normalized linear combination of these eigenfunctions will also correspond to the eigenvalue En.

E n ẳ hCIjHjCIi where jC I i ẳX n i a i jc i i andjCIiis normalized anda 2 1ỵa 2 2ỵ ỵa 2 n ẳ1 Thus hCIjCIi ẳX n iẳ1 ja i j 2 hc i jc i i ẳ1:

There arenpossible linear orthogonal combinations.

A symmetry operation of the system acting on a set of degenerate eigenfunctions takes them into a different linear orthogonal combination of the degenerate eigenfunctions:

CAẳ a11jc 1 i þa12jc 2 i þ þa1 njc n i a21jc 1 i þa22jc 2 i þ þa2 njc n i

an 1jc1i þan 2jc2i þ þannjc n i

In quantum mechanical systems, when all symmetry operations leave a specific eigenfunction unchanged, aside from a phase factor, that eigenfunction behaves as a nondegenerate solution of the Schrödinger equation Conversely, if certain symmetry operations generate new linearly independent eigenfunctions from an existing one, these functions are treated as part of a degenerate set of solutions to the Schrödinger equation.

The eigenfunctions associated with the same eigenvalue in a quantum mechanical system establish a basis for an irreducible representation of the corresponding group The dimension of this irreducible representation is equivalent to the degeneracy of the eigenvalue.

The eigenfunctions of the operator Ei, denoted as HAjc and Ajc, demonstrate that if Ei is nondegenerate and the eigenfunctions are normalized, Ajc equals 1 By applying all symmetry operations of the group, a one-dimensional irreducible representation is formed, characterized by matrix elements and characters of 1 This representation effectively corresponds to the energy state of the system linked to the specific eigenvalue of the eigenfunction In the case of a degenerate state, a similar process yields an irreducible representation with a dimension that matches the degeneracy of the state it describes.

Consider the example of a system withC3vsymmetry described inSect 2.3.1.

A quantum mechanical system exhibiting this symmetry features a nondegenerate eigenfunction that corresponds to the A1 irreducible representation, remaining unchanged under all symmetry operations Additionally, there exists another nondegenerate eigenfunction that remains invariant under E and C3 operations but changes sign under s class operations, serving as the basis for the A2 irreducible representation Furthermore, two degenerate eigenfunctions form the basis for the two-dimensional E irreducible representation, collectively designated as G3.

; whereAis a symmetry operator inC3v This leads to

Ac1ẳG3ðAị 11 c 1 ỵG3ðAị 21 c 2 ;

Ac 2 ẳG3ðAị 12 c 1 ỵG3ðAị 22 c 2 :

This shows thatc1transforms like the first column of the transformation matrix of the symmetry operator whilec2transforms like the second column.

When spin–orbit interaction is important, the total wavefunction describing the system is the product of spatial and spin functions:

In quantum mechanics, the spin angular momentum of a system is represented by double-valued representations, as discussed in Chapter 4 Any physical process that interacts with the system can be described using a quantum mechanical operator, which transforms according to one of the irreducible representations of the group The transformation of a specific operator \( O_n \) from a set of \( n \) operators is expressed accordingly.

O n j G n ðAị ji ; (2.23) whereAis an element of the group of the Hamiltonian andG n is a representation of this group.

The physical processes can lead to transitions between eigenstates or the splitting of degenerate energy states into lower degeneracies Group theory techniques, such as direct products and representation decompositions, are essential for determining the qualitative features of these effects and evaluating matrix elements In quantum mechanics, the evaluation of matrix elements, denoted as ⟨f|O|i⟩, involves the initial and final states of the system, with O representing the physical operator For the matrix element to be nonzero, the integrand must exhibit symmetry, as it represents an integral over all space Instead of calculating the full mathematical expression, we can express it in terms of group theory representations, where ⟨f|O|i⟩ is nonzero if the representation conditions are met.

The matrix element is nonzero, indicating an allowed transition, if the decomposition of the triple direct product representation \( G \times G \times G \) includes the totally symmetric \( A_{1g} \) representation Conversely, if \( A_{1g} \) is absent, the matrix element is zero, signifying a forbidden transition This can be rephrased by noting that \( A_{1g} \) appears solely in the decomposition of a representation when combined with itself Therefore, for a nonzero matrix element, the direct product representation of the initial and final states must encompass the irreducible representation of the operator responsible for the transition.

Transitions Between Vibrational Energy Levels

Tiêu đề	Symmetry, Group Theory, and the Physical Properties of Crystals
Tác giả	Richard C. Powell
Người hướng dẫn	Christian Caron, Managing Editor
Trường học	University of Arizona
Chuyên ngành	Physics
Thể loại	Lecture Notes
Năm xuất bản	2010
Thành phố	Tucson

Định dạng
Số trang	240
Dung lượng	1,62 MB