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Understanding Crystal Symmetry in Physics

Crystal symmetry plays a crucial role in determining physical properties of crystals, impacting phenomena like electric displacement and optical properties. Group theory concepts outline the symmetries of crystals, where each crystalline structure possesses unique geometric operations that leave the crystal unchanged. This symmetry is essential for understanding optical and photoemission spectra, enabling the determination of the one-electron potential through empirical or self-consistent approaches. Crystal symmetry relates to lattice symmetry and Bravais lattices, influencing material behavior based on the crystallographic point group and space group.

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Understanding Crystal Symmetry in Physics

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  1. PHYS 5560 Lecture 3

  2. How to find the One Electron Potential U(r)? • Empirical Approach – Parametrize U(r) in as few constants as possible – Determine these constants by comparison with experiment (optical and photoemission spectra) • Self-consistent Approach – Start with a trial U1(r), calculate electron wavefunction, density and potential U2(r) – If U2(r)≠U1(r), use U2(r) instead and repeat until Un(r) =Un+1(r)

  3. A very brief introductions on crystal symmetry and group Crystal symmetry: Under certain geometrical operations, the crystal is unchanged, therefore, the final and initial state are the same in physics. (Still, when we measure, we really have error tolerance, therefore, crystal symmetry is largely depends on our error tolerance.) Lattice symmetry: symmetry of Bravais lattice. (For each primitive unit cell, we shrink it into a lattice point of Bravais lattice, that has the highest symmetry) Crystal symmetry is very important to the physical properties of crystals. For example, electric displacement vector: For cubic lattice, the permittivity can be regarded as a scalar, because cubic lattice is isotropic along all three directions. However, for hexagon lattice, becomes birefringence D  =   E 

  4. Basic concept of group (For complete reading, please refer to Peter Yu’s book or “Group Theory: Application to the Physics of Condensed Matter” ) Definition: a set (G={E…,P,…}) with operational elements that satisfy the following constraints: (Note that the operation does not have to be a mathematical multiplication) (1) closure closure, for any A,BG, their product, AxB G (2) There exists a unique unit element,E(identity P, PxE=ExP=P identity)，for all elements, inverse element, P-1, (3) For any element P, there exists a unique inverse PxP-1=P-1xP=E; (4) Associativity Associativity, Ax(BxC)=(AxB)xC The definition of a group does not require that for all elements AxB=BxA. If this additional condition holds, the group is commutative abelian group abelian group. Otherwise, it is a nonabelian nonabelian group commutative and called an group The number of elements is called order order. One question, does additive operations form a group for integers? How about multiplication of real numbers?

  5. ：For any element S and R in G，R‘=SRS-1and R are called Conjugate： conjugate. Class: in group G, all conjugated elements form a class. Subgroup: If H is a subgroup of G，it contains the same operations as G. Invariant Subgroup: If H is a subgroup of G，For any elements aG, bH, if aba-1H, then H is an invariant subgroup of G, or normal subgroup. For example: {additive group of integers} is {additive group of real number}’s invariant subgroup.

  6. Coset Coset: : If H element of H, left element of H, left coset ，if if g is an element of g is an element of G G, coset of H: of H: { {gh gh}; }; , h is each h is each If H is a subgroup of G is a subgroup of G， Similarly, if h is an element of H and g is every element of G, Similarly, if h is an element of H and g is every element of G, Right Right coset coset of H: of H: {hg}. {hg}. factor group factor group: : If H If H is an invariant subgroup of G is an invariant subgroup of G， coset coset, if h is each element of H, G/H = { , if h is each element of H, G/H = {gh {G {G - -> G/H} > G/H} sending each element sending each element g ，g is an element of G, the left g is an element of G, the left gh| | g g G G}. }. (G modulo H) maps (G modulo H) maps g to to gH gH is a homomorphism is a homomorphism. .

  7. Equilateral triangle, D Equilateral triangle, D3 3 group{ group{e,a,b,c,d,f e,a,b,c,d,f} } e: self a: rotation around axis 1 for   b: rotation around axis 2 for   c: rotation around axis 3 for   d: rotation around axis Z for 2 /3 f: rotation around axis Z for 4 /3 Multiplication table of D Multiplication table of D3 3

  8. D D3 3group{ group{e,a,b,c,d,f e,a,b,c,d,f}, three classes {e {e} } class class， ， }, three classes： ： a a class{ class{a,b,c a,b,c}, },和 和 d d class { class {d,f d,f} }。 。 H={ H={e,d,f e,d,f} } is an invariant subgroup is an invariant subgroup， aH aH={ ={a,b,c a,b,c} } ， aH aH forms a factor group D forms a factor group D3 3/H /H symmetry group: symmetry operations form a group. Atoms & Molecules have Rotational Symmetries (Reflections are regarded as improper rotations)

  9. Point Group Operation 1. (rotation) C axis(2 /n) n-fold axis For example Cz( /2)(x,y,z)=(y,-x,z) 2. inversion, I(x,y,z) (-x,-y,-z) ，(x,y,z) (y,x,z) 3. reflection for a plane x=y，

  10. Example of a Point Group

  11. Point group: Only contains rotational operations with a fixed atom. With crystallographic restriction, there are 32 point groups. Space group A combinations of the 32 crystallographic point groups with the 14 Bravais lattices, There are totally 230 space groups. Translational group is an invariant subgroup of space group.

  12. Nonsymmorphic Space Groups • Space groups are divided into two kinds: – Operations in Symmorphic groups=a point group operation+a translation where both the point group operation and translation are symmetry operations of the crystal – In nonsymmorphic groups there is at least one operation which consists of rotation and translation such that rotation or translation separately are not symmetry operations (screw axis, glide plane) • Of 230 space groups only 73 are symmorphic

  13. One Nonsymmorphic Symmetry Operation: Screw Axis -rotation about the axis+ translation along the axis

  14. Nonsymmorphic Symmetry Operation: Glide plane -reflection about the plane+translation parallel to the plane

  15. Example of a Symmorphic Group: Td2 (Zincblende) • Point Group Symmetry Operations:

  16. Example of Nonsymmorphic Space Group:Oh7(Diamond) •Consider reflection in [100] plane •This is not a symmetry operation since there are no atom on the other side •However, a translation by (a/4)(1,1,1) brings the crystal back to itself •This operation is known as a glide and the [100] plane is a glide plane •Since there are 3 glide planes they result in the symmetry operation: i’ =inversion followed by translation by (a/4)(1,1,1)

  17. Another Example of Nonsymmorphic Space Group:C6v4 GaN (Wurtzite Structure) Top View showing the glide plane AA’

  18. Relation between Space Group & Point Group • For symmorphic space groups every symmetry operation can be decomposed into the product of two operations: a point group operation and a translation. Thus the group of point group symmetry operations is also a subgroup of the space group. The symmorphic space group can be simply regarded as the “product” of two groups: a point group and the group of all translations.

  19. Relation between Space Group & Point Group For non-symmorphic groups the group of point group symmetry operations is not a group. There will be symmetry operations of the space group such as the point group operation when separated from the translation operation is not a symmetry operation. We can “capture” these operations containing translation to form a point group by constructing a homomorphism between the factor group G/T (where G is the space group while T is simply the set of translation operations) and R which contains the set of point group operations.

  20. Matrix representation of groups： A set of such transformation matrices corresponding to a group of symmetry operation is also a group, which is said to form a representation of the symmetry group. ： Rotation along z axis, det| R| =1 Inversion, det| R| =-1

  21. Representations and Basis Functions Representations and Basis Functions • Choose some function f(x,y,z), then we can generate a set of functions {fi} by applying the symmetry operator Oi of the group to f(x,y,z), so that fi=Oi[f]. Due to the closure requirement of the group [ ] i j ji j  O f f a a matrices • Symmetry operations in a group can be represented by the way they transform a set of functions into each other. The functions {fi} is said toform a set of basis functions. • These transformation matrices contain information on the symmetry of the functions and is known as representations of the group. form a square matrix,  a transformation • ji

  22. These transformation matrices are not unique since they depend on the functions chosen. However, the sums of their diagonal elements (trace) are the same for functions of similar symmetry. The trace is known as the character of the functions corresponding to the various symmetry operations. Instead of working with the transformation matrices it is often more intuitive to work with an appropriate choice of functions (basis functions) since the character can be obtained with any set of basis functions; e.g. the three functions: {x,y, and z} can be chosen as the basis functions representing all p states with the angular momentum l=1.

  23. Irreducible Representations When the transformation matrices for all the symmetry operations for certain functions can be reduced to smaller square matrices, then the representation is said to be reducible. Basis functions with distinct symmetry gives rise to different irreducible representations Elements in the group can be divided into classes. aT=Ta, a any of element of the group, a set of group elements, T belong to a class. •Elements belonging to the same class have the same character. •For a given group the number of irreducible representations is equal to the number of classes in the group.

  24. Character Table • A “square” table showing the characters of all the irreducible representations of a group is known as the CHARACTER TABLE.

  25. Example:

  26. Construction of the Character Table from a Group • Instead of calculating the characters from the trace of transformation matrices, the character table can be constructed by using the following orthogonality relations:          * C C N h i k j k k ij k iCk   *iCl   (h / Nl)kl i    iC Character of Ckin Rirepresentation is denoted as h: order of group; Nk # of elements in a class k

  27. Example of construction of Character Table of Tdby Inspection • Order of Group=total number of elements=h=24 • Number of classes=5. They are: • {E}, {3C2}, {6S4}, {6s}, {8C3} • By applying the orthogonality relation to the identity class we must have:   E  i 2 i    24 h • Since the number of irreducible representations is equal to the number of classes, 24=sum of 5 squares!

  28. Example of construction of Character Table of Td by Inspection • It would not take long to find out that the following combination is the only one that works: • 12+12+22+32+32=24 • In this way we obtain the first column and the first row of the character table of Td

  29. Example of construction of Character Table of Tdby Inspection • To obtain the other characters by inspection requires some practice but the trick is the same. For example consider the characters corresponding to {6s} or {6S4}. • From the orthogonality relation the sum of the characters squared =24/6=4. This implies that 4 characters are equal to =1 or -1 while 1 of the characters have to be zero. From the other orthogonality relation we find that the sum of the products of the characters of {E} and {6S4} or {6s} has to be zero. This implies that the characters of the 3D representations T1and T2has to cancel each other. In other words if one is +1 the other has to be -1. Similarly one finds that the characters for the A1 and A2 have to cancel each other leaving the character for E to be zero. From these arguments we arrive at the two columns corresponding to {6s} and {6S4}.

  30. Example of construction of Character Table of Td by Inspection • Next we apply the orthogonality relation to {8C3}. The sum of the characters square is equal to 24/8=3 so there are only 3 characters of either 1 or -1 and two characters equal to 0. • If we assume that the character of one of the 3D representation is non-zero, say equal to -1, then we have to assume that the character of E has to be 1. This choice will not satisfy the orthogonality relation between {8C3} and {6s}. • The other choice we have is to make both characters of the 3D representations equal to zero. Then remaining choices for the other characters has to be :  (E)=-1 and (A2)=-1 so that the orthogonality relation is satisfied for the characters of {E} and {8C3}. • By using the same arguments one can eventually derive the character table of Td.

  31. Three commonly used Notations for the Irreducible Representations of Td

  32. Character Table for Ohgroup

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