discrete symmetry groups in solid state physics · 2017-02-21 · symmetries, groups and algebras 2...

Discrete symmetry

groups in solid state physics

Joakim Hirvonen GrutzeliusKarlstad University 16th March 2004

Department of Engineeringsciences, Physics and MathematicsSymmetries Groups and Algebras 5p

Examinator: Prof Jurgen Fuchs

Abstract

The main topic of this project are the discrete symmetry groups that canbe used in applications in solid state physics. These groups and their extensionare going to be discussed and visualized in stereograms. For convenience mostof the illustrations are based on the Cyclic groups Cn.

At the end of the project there is a table of several holosymmetric pointgroups which define 7 crystal systems based on the primitive ones. There isalso a short discussion about the most common used point groups and theirrelations to Lie groups.

Symmetries, Groups and Algebras 1

Contents

1 Discrete Symmetry Groups 2

2 Point Groups 22.1 Symmetry Elements . . . . . . . . . . . . . . . . . . . . 22.2 Proper Point Groups . . . . . . . . . . . . . . . . . . . . 42.3 Improper Point Groups . . . . . . . . . . . . . . . . . . . 6

3 Colour Groups and Magnetic Groups 8

4 Double Groups 10

5 Lattices, the Translation Group and Space Group 135.1 Normal Space Groups . . . . . . . . . . . . . . . . . . . . 135.2 Colour and Magnetic Space Groups . . . . . . . . . . . . 165.3 Double Space Groups . . . . . . . . . . . . . . . . . . . . 17

A Groups and Lie groups 18

B Table of crystal systems and Bravais lattices 19

Joakim Hirvonen Grutzelius Student


1 Discrete Symmetry Groups

Those discrete groups which play the central role in solid state physicsare the point groups and their extensions (double, colour groups), thetranslation groups and the combination of both (the space groups).These groups and the meaning of their elements are discussed in thefollowing sections. Apart from these, the symmetric (permutations)groups are important in physical systems, but are not going to be dis-cussed in this project.

2 Point Groups

2.1 Symmetry Elements

To understand the concepts of symmetry it can be nice to keep in mindsome basic symmetry operations. We will first consider those symmetryoperations that keep one point fixed and leave the distance between twopoints of the space (in general R3) unchanged.The points of the space are described in a Cartesian coordinate systemwhere the origin is chosen to be the fixed point.The first thing that we will do is to define symmetry elements for somebasic symmetry operations.So let us first look at the simple rotation about an axis through the fixedpoint. The symmetry element that correspond to a rotation throughan angle 2π/n we will denote cn. The order of the element is n and theaxis is an n-fold one, so that:

cnn = e or cm

n = cn+mn (1)

We also have to specify the direction of the rotational axis. For exampleif the n-fold rotation axis point in the z-direction we write:

cn(z) = cnz (2)

If we now consider a reflection in a plane, the corresponding symmetryelement have the order 2 and is denoted by σ. So we have:

σ2 = e or σ−1 = σ (3)

Of course, we also need to declare the orientation of the mirror planeand we will do this by specifying a mirror plane throw the directionof the normal to the plane. For example, if we have a mirror planeparallell to the xy-plane we will write it as:

σ(z) = σz (4)

We can easily visualize these two symmetry operations but first wechoose a convention as follows, we say that the coordinates of a point,or position vector, are always taken with respect to a fixed coordinate



system and then that the symmetry elements are operators in R3. Ifwe studie figure 1 (below) we see a rotation d of an point P with posi-tion vector x = (x, y, z). Which under the rotation is transformed toanother point P ′ with position vector x ′ = (x′, y′, z′).

Figure 1: Rotation of a vector x with position P onto position P ′ and reflection inthe yz-plane (P onto P ′′)

We describe this procedur by:

x ′ = dx ; d = c (ϕ) (5)

If we rotate the vector through an angle ϕ, counterclockwise, and therotation axis point in the z-direction we can describe it as follows: x′

y′

z′

=

cos ϕ − sin ϕ 0sin ϕ cos ϕ 0

0 0 1

xyz

(6)

In figure 1 (above) we also can see a reflection in the yz-plane whichcorrespond to the matrix:

σ(x) = σx = σv =

−1 0 00 1 00 0 0

(7)

From these basic symmetry elements we can derive a couple of otherelements. For example if we have an reflection in three orthogonalplanes (x = 0, y = 0, z = 0) and take the product of these we will getthe (space) inversion:

i = σ(x) · σ(y) · σ(z) ; i = −δik (8)

If we simultaneously have a rotation and a reflection we call this amirror rotation and describe it with a rotation through an angle 2π/nabout an n-fold axis combined with a reflection σh at a plane perpen-dicular to the rotation axis:

sn = cn · σh = σh · cn , smn =

{cmn m even

cmn · σh m odd

(9)



then we getsn

n = e for n evensn

n = σh , s2nn = e for n odd

(10)

The order of sn, is therefore 2n for odd n. Instead of a mirror rotationoften the roto-inversion in is used:

in = cn · i = i · cn , imn =

{cmn m even

cmn · i m odd

(11)

with a relation corresponding to (10). The operations (9) and (11) arenot independent as we can see in figure 2 (below).

Figure 2: Illustrations of S3 (left) and J3 with the help of stereograms. ©: Pointsa distance z above the plane of the paper.

⊗: Points at distance −z (below the

plane).

In particular we have:

i1 = i = s2 , i2 = s1 = σh

im4 = s4−m4 , im3 = s6−m

6 , im6 = s6−m3

(12)

So far we have distinguished between two different types of reflections,in principle, there are three kinds of reflections:

σh: reflections in a plane perpendicular to the n-fold main rotation axis(h:horizontal).

σv: reflections in planes containing the main axis (v:vertical).σd: reflections in planes containing the main axis and bisecting the

angle between neighbouring additional axes (d:diagonal).

2.2 Proper Point Groups

A rotation group is a group of symmetry operations that leave onegiven point and all the angles and distances in Euclidean space (R3)unchanged. If there are only rotations in such a group we will call ita proper rotation group. This group is isomorphic to a subgroup ofS O(3) = R(3) of all orthogonal (3× 3) matrices with determinant 1.If we take the direct product of a proper group with the inversion groupZ2 = {e, i} we will get a improper rotation group, which is isomorphicto a subgroup O(3) = Z2 ×S O(3) of all orthogonal matrices. Therealso exist a homomorphism f : O(3) → S O(3) with the kernel NK =



Z2 = {e, i}; the quotient group of O(3) with respect to Z2 is O(3)/Z2∼=

S O(3).Proper and improper point groups are for example finite subgroups

of S O(3) and O(3), respectively. We will begin to consider properpoint groups of finite order which describe the symmetries of moleculesand the unit cells in crystals.

i) We have the cyclic (rotation) group Cn (= n)(also denoted Zn)It contains n elements. The elements of this group are just powersof the generating element cn. We write it as follows:

Cn = {cmn |m = 0, ..., n− 1}. (13)

These groups describe arrangements with an n-fold axis and theyare Abelian so that every element forms a class of its own.

ii) The dihedral groups Dn (= n22 for even n, = n2 for odd n).These groups describe symmetry arrangements with one n-foldmain axis and in addition n2-fold axes perpendicular to the mainaxis at angles of π/n to each other.

Dn = {cmn ; c2, c′2, ...|cn

n = (cnc2)2 = e}. (14)

A regular prism with n surfaces possesses this symmetry. Theorder of Dn is 2n, the generating system is {cn, c2 ⊥ cn}. Thegroup is not abelian for n ≥ 3; the number of classes is equal to(n + 6)/2 for even n, and (n + 3)/2 for odd n.

Apart from these groups, there are only three additional ones with morethan one main axis. These are as follows:

iii) The tetrahedral group T (= 23)This group possesses four 3-fold axes through a vertex and thecentre of the opposing face of a tetrahedron, and three 2-fold axeswhich are perpendicular to each other and pass through the centresof opposing edges.

T = {c3, c′3, ...; c2, c′2, c′′2|c33 = c3

3 = (c3c2)3 = e}. (15)

The order is 12, the number of classes 4 and the number of gener-ators 2 (c3 and c2).

iv) The octahedral group O (= 432)Is described by the axes of a regular cube or an octrahedron. Thereare three 4-fold axes which are perpendicular to each other, six 2-fold axes and four 3-fold axes. The order is 24 and the number ofclasses are 5. As a system of generators we can choose:

O : c44 = c2

2d = (c4c2d)3 = e ; c4c2d = c3 or (16)

O : c44 = c′3

3 = (c4c′3)

2 = e ; c4c′3 = c2d . (17)



The elements have the following meanings: c4, rotation about z-axis; c3, rotation about the cube diagonal −x = y = −z; c2d,rotation about the face diagonal x = z, y = 0. The tetrahedralgroup is a subgroup to the octahedral group with index 2. Thiscan be seen by embedding a tetrahedron in a cube so that thevertices of the tetrahedron coincide with the vertices of the cube.

v) The icosahedral group Y (= 532)is the symmetry group of the regular icosahedron and dodecahe-dron. It is discussed in physical problems with fullerenes and inthe theory of quasicrystals. The group consists of 6 fivefold, 10threefold and 15 twofold axes, altogether we have 60 elements in5 classes. There are again two generators,

Y : c55 = c2

2 = (c5c2)3 = e, (18)

for which the directions of the axes have to be chosen in an ap-propriate way.

2.3 Improper Point Groups

We can easily derive the improper point groups from the proper ones.We will use two different methods which together will give us all possibleimproper point groups.1) By the formation of outer direct products with the inversion groupZ2 = {e, i}.2) By decomposition of the point group G into cosets with respect toan invariant subgroup of index 2, G = N +iaN , and by the formationof G = N + aN with the inversion i.For example when using method 1 we get:

1. a) From Cn we obtain Cnh (even n) and S2n (odd n).

b) From Dn we obtain correspondingly Dnh (even n) and Snd

(odd n).

c) From T , O, Y we obtain Th, Oh, Yh.

And if we use method 2:

2. a) From C2n with invariant subgroup Cn we obtain S2n (even n)and Cnh (odd n).

b) From Dn with invariant subgroup T we obtain correspond-ingly Cnv.

c) From D2n with invariant subgroup Dn we obtain correspond-ingly Dnd (even n) and Dnh (odd n).

d) From O with invariant subgroup T we obtain Td.

e) T and Y do not possess an invariant subgroup of index 2.



If we studie these exampeles we can see that some of them is identically,not only isomorphic, see for example 1a) and 2a) or 1b) and 2c). Thisalso holds for mirror rotation and roto-inversion groups. Thus we have:

Ci = I1 := I = S2 , I2 = S1 ,I4 = S4 , I3 = S6 , I6 = S3 ,

(19)

andCs = C1h , C3 = S6 , C3h = S3 (20)

Isomorphic groups are

Dn∼= Cnv , Ci

∼= Cs∼= S2 , S3

∼= C4 ,

D2∼= C2h , C6

∼= C3h∼= C3i , D4

∼= D2d , (21)

D6∼= D3h

∼= D3d , Td∼= O

Sometimes it is useful to consider the orthogonal rotation groups O (n)and S O (n), with n = 2, 3, as the limits of the discrete groups forthe symmetry of a cylinder and sphere, respectively. As we have seenearlier the point groups can be illustrated in stereograms. See figure 3(below) for examples of C3v and D3. These groups are isomorphic butthey describe different symmetries.

Figure 3: Stereograms for C3v (left) and D3 (right) symmetry. For C3v all the pointslie in one plane.



3 Colour Groups and Magnetic Groups

When we have physical objects described by the symmetry elementsof point groups they do not have an internal physical structure. Butthese object can possess internal degrees of freedom besides the externalgeometrical degrees of freedom. In the following discussion the innerdegrees of freedom will be restricted into two values. We look uponthese values as two colours, black and white, or we can think of itat as two possible directions of the magnetic moment (spin parallelor antiparallel to a given direction). A change of such a value wewill describe by the operator r. When we act with the operator r ona symmetry group we enlarge it and we obtain new groups besidesthe ordinary geometrical point groups. We can characterize these asfollows, where r2 = e:

I) Colour groups of the Ist kindThese groups describe objects with a definite colour e.g. white,illustrated in figure 4 (below). The operator r is not a symmetryelement of the system because it would change the colour intoblack, so it follows that this group is equal to the geometricalgroup.

MI = G , r /∈ MI (22)

The number of possible MI is equal to the number of point groupsG .

Figure 4: Colour point group of the first kind illustrated for C I3v.

II) Colour groups of the IInd kindThese are also denoted grey or major groups. They can be de-scribed by the decomposition into cosets.

MII = G + rG ; r ∈ MII (23)

In this case G is an invariant subgroup of MII with index 2 and ris an element of MII and the order of MII is 2g. If we look on theobjects of MII (see figure 5 below) they simultaneously possessboth colours (as tought in kindergarden, if you mix two coloursyou get another one, in this case grey) so a change from black towhite or vice versa does not have any consequence. It follows thatwhen operating with the operator r it leaves the internal degree of



freedom unchanged. As r commutes with all elements of G , MII

can be written as a direct product:

MII = G × {e, r} (24)

Figure 5: Colour point group of the second kind illustrated for C II3v .

III) Colour groups of the IIIrd kindThey do not possess r as a separate element, but they containelements that are coupled with r.

MIII = N + r (G −N ) = N + ra′N , a′ ∈ G −N (25)

where N is an invariant subgroup of G with index 2. As we cansee in figure 6 below, the objects of this group are black and whiteand equal in number.

Figure 6: Colour point group of the third kind illustrated for C III3v .

So there is certain symmetry operations that transforms the sys-tem into itself only with a simultaneous change of colour. We havethen that the order of MIII is that of G and then since r commuteswith all the elements of G we have:

(ar) bi (ar)−1 = abia−1 = bj

(ar) bir (ar)−1 = abira−1 = bjr , a, bi, bj ∈ G (26)

this means that the classes of MIII correspond to those of G ; theyhave either only elements of N or of r ·(G −N ). A point group Gmay have different invariant subgroups of index 2. Therefore thenumber of MIII groups is larger than that of the common pointgroups.



So when are these group of interest? The fact is that the colour groupsare important when they are used as magnetic groups. Then r becomesthe time reversal operator ϑ which reverses the direction of a physicalevent, in particular, the direction of a current. And we know that amagnetic moment correspond to an electric current and behaves thesame way.

4 Double Groups

In many physical systems the objects (atoms, electrons, nuclei) possessa spin which we of course have to take into account. A case of specialinterest is when the objects possess a spin 1/2 (fermions). Now we canask us why this is of interest and the answer is that this will lead usbeyond physics with integer quantum numbers for the angular momen-tum. If we think of nonrelativstic quantum mechanics where the spinis described by enlarging the wave function to a spinor wave functionϕv containing two components (v = 1, 2). The spinor is characterizedby its behavior under rotations. If we first look upon a general rotationof a vector in R3 we have:

x′ = c (α, β, γ) · x, (27)

with c (α, β, γ) = cos α cos β cos γ − sinα sin γ − sinα cos β cos γ − cos α sin γ sinβ cos γcos α cos β sin γ + sinα cos γ − sinα cos β sin γ + cos α cos γ sinβ sin γ

− cos α sinβ sinα sinβ cos β

Where the rotation is specified by the Euler angles. In a rotation likethis the transformation of the two component spinor becomes:

ϕ′ = D1/2 (c (α, β, γ) · ϕ) , (28)

with

D1/2 (c (α, β, γ)) =

(expi(α+γ)/2 cos β exp−i(α−γ)/2 sin β− expi(α−γ)/2 sin β exp−i(α+γ)/2 cos β

)This (2×2) matrix reproduces itself only under a rotation by 4π, notby 2π, and therefore ϕ′=-ϕ after a rotation by 2π, but ϕ′=ϕ after ro-tation by 4π. To every physical rotation c (α, β, γ) belong two matricesD1/2; they differ in the rotation angle by 2π: α and 2π + α, etc. Thenumber of parameters (α, β, γ) is equal in both cases, i.e. the matricesonly differ in sign. The matrices (22) form the real special orthogonalgroup S O (3) while the matrices (23) forms the special unitary groupS U (2). These matrices obey DD+ = 1 and detD = 1.As we just discussed it follows that two elements of S U (2) corre-spond to one element of S O (3). And therefore we have a two-to-onehomomorphism S U (2) → S O (3) in which the matrices:(

1 00 1

)and

(−1 00 −1

)(29)



correspond to the matrix 1 0 00 1 00 0 1

∈ S O (3) (30)

i.e. (24) is the kernel of the homomorphism. The same relation thatS U (2) has to S O (3) holds for a double groups relation to a pointgroup. This means that if G is a subgroup of S O (3) of order g, thenthe double group G D (order2g) belonging to G has a group table thatcan be derived from the corresponding matrices of S U (2). The doublegroup G D is a subgroup of S U (2). We can describe the double groupby adding a new element c0. If we say that c (α) is the rotation throughan angle α, then the elements of the double group have to obey:

c (α + 2π) = c (α) · c (2π) 6= c (α) , (31)

c (α + 4π) = c (α) · c (4π) = c (α) , (32)

withc (2π) = c0 , c2

0 = e (33)

As we see above that a rotation through 2π is not identical with e. IfG = {e, a1, a2, . . . } ⊂ S O (3) the corresponding double group is givenby:

Cn = {e, a1, a2, . . . ; c0, c0a1, c0a2, . . . }, (34)

which is isomorphic to a subgroup of S U (2). Note that G itself is nota subgroup of G D because the elements of G D that correspond to theelements of G do not form a closed set in G D (e.g. c2

2 = c0 /∈ G ). Forexample, if

G = Cn = {e, cn, . . . , cn−1n } and c0 = c (2π) = cn

n,

thenG D = C D

n = {e, cn, . . . , cn−1n , cn

n, . . . , c2n−1n },

which is a cyclic group of order 2n. As we can see from above, Cn isnot a subgroup of C D

n since cnn = c0 /∈ Cn in C D

n .When we have determined the double groups of the proper point

groups Cn, Dn, T , O, Y , the improper groups follows from section 2.3.The elements ai and c0ai ∈ GD in general belong to different classes;

i.e. if Ki is a class of G , then the double group has the correspondingclasses Ki and KD

i . However there are exceptions:

i) If c2 is a rotation through an angle π, then c2 and c0c2 belongto one class iff there exists in G either another twofold rotationperpendicular to the axis of c2 or a mirror plane that contains theaxis of c2.

ii) If σ is a reflection, then σ and c0σ belong to one class iff thereexists in G either another mirror plane perpendicular to the firstone or a twofold rotation with its axis in the plane of σ.



Both statements follow from the noncommutativity of the correspond-ing elements. For example if we look at the groups:

D2 = {e, c2x, c2y, c2z} and C2v = {e, c2z, σx, σy},

which are abelian with 4 classes each. The double groups are:

DD2 = {e, c2x, c2y, c2z, c0, c0c2x, c0c2y, c0c2z}

C D2v = {e, c2z, σx, σy, c0, c0c2z, c0σx, c0σy}

which are not abelian, with the five classes:

{e} , {c0} , {c2x, c0c2x} , {c2y, c0c2y} , {c2z, c0c2z} and

{e} , {c0} , {c2z, c0c2z} , {σx, c0σx} , {σy, c0σy}We obtain the decomposition into classes by conjugation, e.g.

c−12x c2zc2x = c0c2xc2zc2x = c0c2z , i.e. c2z and c0c2z

belong to one class. For other groups we have, e.g.

c−12x c3zc2x = c0c2xc3zc2x = c0c

23z

c−12x c0c3zc2x = c0c2xc0c3zc2x = c2

3z

In this way we find that in these double groups {c3z, c0c23z} and

{c23z, c0c3z} each form a class but not the same way as in simple point

groups, {c3z, c23z}. Similar relations hold for other elements and double

groups. In applications this has to be observed very strictly.



5 Lattices, the Translation Group and Space Group

5.1 Normal Space Groups

The physical objects (atoms, ions, molecules) of materials in the solidphase are often arranged in regular periodic structures, these structureswe call lattices or Bravais lattices. If we look upon a lattice structureas infinite we call it an ideal lattice structure. The ideal lattice struc-ture can be described by a periodic repetition of identical unit cells.We describe every unit cells by three non-coplanar basis vectors a(i).According to the lengths of the a(i) and the angles between them wewill get different lattice structures. By this it follows that the differ-ent structures will have different symmetry operations. Not all pointgroups symmetries are compatible with a lattice structure. If we wantto have an agreement between them we need to fill the hole space withpolyhedra or polygons. A plane for example has to be covered with equi-lateral polygons with m vertices. A point group symmetry demands ann-fold rotation axis through one vertex, which then transforms the setsof polygons into itself. We have that an integer multiple of the anglebetween two edges in the equilateral m-polygon, ϕ = (m− 2) π/m, hasto be equal to 2π, so we get the following:

nϕ = m (m− 2) π/m = 2π or n = 2m/ (m− 2) (35)

Since m and n have to be integers we have the following:

m = ∞, 6, 4, 3, 2 , n = 2, 3, 4, 6,∞ , (36)

where we have an n-fold axes for n. In an infinite lattice we have only2-,3-,4- and 6-fold axes (n = ∞ means isotropy about an axis). In R3

a corresponding consideration provides the same result. We also havethe mirror planes as possible symmetry elements.

Table 1 in appendix C shows the possible lattice structures and themaximal point groups symmetries compatible with them. The latticestructures implies that the inversion is a necessary element. If we startwith the most primitive type of lattice (P and R in table 1) furthertypes can be derived by implanting ”centered” lattice points (C,F,I intable 1). The centered lattices have the same point group symmetry asthe primitive ones. There are 7 of these holosymmetric point groups,which defines 7 crystal systems.

The positions of all unit cell or all the vertices of a lattice are givenby:

Rh =3∑

i=1

hia(i) = h1a

(1) + h2a(2) + h3a

(3) , hi ∈ Z (37)

When we assign an identical set of points, a basis, to every unit cell,we obtain the ideal crystal structure. Those symmetry operations thattransforms the crystal structure into itself form the space group R. The



space group comprises translations (displacements of the atoms relativeto the lattice structure) and point symmetry operations. We can alsohave combinations of the point symmetry operations and translations.One of the combinated elements are glide planes which contains a si-multaneous reflection ata plane and translation parallel to the plane.Another one is the screw axes which is a simultaneous rotation through2π/n and translation by ma/n , m = 1, 2, . . . , n− 1 , a:basis transla-tion. These operations is illustrated in figure 7 (below).

Figure 7: Screw axis 4m. The fractions give the height above the plane of the paperin units of the screw period. The elements 41 and 43 are enantiomorpic (left- andright-handed screw axes). C III

3v .

The translation connected with a glide plane is always a/2, since σ = 1.Screw operations always have the form nm. Some of the screw axis wecall enantiomorphic which means that they only differ in the sense ofthe screw (right-left). We can write the elements of the space group as{d|t}, where d means a point symmetry element and t a translation.The space group elements are operators in R3-space which act on points(or positions vectors) of the space:

x′ = {d|t} · x := dx + t (38)

With two successive operations (33) we obtain the rule of multiplica-tion:

{d2|t2} · {d1|t1} = {d2d1|t2 + d2t1} (39)

{d|t}−1 = {d−1| − d−1t} defines the inverse element (40)

{e|t} is a pure translation by t (41)

{d|0} is a pure rotation or reflection (42)

{e|0} is the identity element (43)

If we have a translation that transform an empty lattice, that do notcontains particles, into itself, we denote it as {e|Rh}. These transla-tions form the discrete, countably infinite translation group T of thelattice or crystal. When there are only one atom in the unit cell wehave a Bravais crystal. If we choose basis vectors in such a way thatevery lattice point can be reached by a multiple of the basis we define



this cell as the primitive unit cell. We can use another method to ob-tain the unit cell which we will call the Wigner-Seitz cell, this methodwill reflect the symmetry of the crystal best. A definite lattice pointRh = 0 is connected with all neighbouring points. Then the planes(r · Rh

)= Rh2

/2 have to be determined, these planes are perpen-dicular to the connecting lines and bisect them. So that the smallestpolyhedron defined by these planes is the Wigner-Seitz cell. It is invari-ant with respect to the holosymmetry of the lattice. We get the volumeof the unit cells throw the scalar triple product Vz = a(1) ·

(a(2) × a(3)

)of the basis vectors.

The translation group T is an Abelian invariant subgroup of thespace group R:

{d|t} · {e|Rh} · {d|t}−1 = {d|t + dRh}{d|t}−1 = {e|dRh} (44)

{e|dRh} is a translation covering all elements {e|Rh} of T when Rh

assumes all possible values.The pure point group elements {d|0} are said to be the point group

G of the lattice. The lattice according to the translation vector Rh

is not changed under any operation d ∈ G . A complete description ofthe space group R also needs a specification of possible nonprimitivetranslations s(d) connected with glide planes and screw axes. Since Tis a invariant subgroup of R, the space group can be decomposed intocosets with respect to T. The quotient group R/T is isomorphic to thepoint group G :

R =∑d∈G

{d|s(d)} · T (45)

If we choose a appropriate coordinate system so that all the s (d) canbe put to zero, then {d|0} ∼= G is a subgroup of R. Since T is aninvariant subgroup of R and that

G ∩ T = {e|0} , {d|0} · T = T · {d|0} , {d|Rh} = {e|Rh} · {d|0} ,

all the assumptions of a semidirect product are satisfied. These spacegroups are called symmorphic groups and can be written as

R = T s G (symmorphic) . (46)

If now s(d) cannot be taken equal to zero, then the space group is anonsymmorphic one.

As an example we will discuss the group D144h = P42/mnm (mnm

are Hermann-Mauguin notation for the symmetry). This is the symme-try group of the paramagnetic phase of some magnetic materials, e.g.MnF2, FeF2, CoF2, NiF2 and MnO2. The permanent magnetic mo-ments of the ions are distributed randomly. The structure is also that ofTiO2, and is therefore called rutile structure. It belongs to the tetrag-onal system, the holosymmetry is D4h with a primitive Bravais lattice.The basis vectors are orthogonal with the lengths a, a, c; and the volumeis a2c. It has a nonprimitive translation s(d) = (a(1) + a(2) + a(3))/2.



5.2 Colour and Magnetic Space Groups

Colour space groups are defined analogously to colour point groups.This holds especially for groups of types I and II.

I) As we discussed in section 3, the colour space groups in this cate-gory describe systems with well-defined colours, that is:

MI = R and r /∈ MI (47)

II) Correspondingly, we have for the grey colour space groups:

MII = R + r ·R and r ∈ MII (48)

For the black-white colour space groups we need to divide into twokinds, IIIa where the elements that changes colours is connected withan element of the point group, while in IIIb it is connected with anelement of the lattice translation.

IIIa) In the first case

MIIIa = N + r(R −N ) r /∈ MIIIa (49)

where N is an invariant subgroup of R with index 2 and R −N does not contain pure lattice translations. The correspondinglattices are the usual Bravais lattices.

IIIb) The element r which changes colour is associated with the lat-tice; we must therefore first define a black-white lattice. We canachieve this, like we did for the point groups, by combining anextra translation {e|t} (or a certain superposition of translations)with an element r that changes colour. When we have appliedsuch a translation twice, the original colours has to be restoredagain. In this way we get the lattices of the type:

TIIIb = T + r{e|t}T (50)

The space group is then

MIIIb = R ′ + r{e|t}R ′ (51)

where R ′ is a space group that does not contain the element {e|t}.These space groups occur in connection with magnetic structures. Asin section 3 the element r that change colour is then the time revereloperator ϑ. Crystals with magnetic properties belong to these spacegroups as follows:

i) Dia- and paramagnetic phases: all MII .

ii) Antiferromagnetic phases: all MI , MIIIa, MIIIb.

iii) Ferro-(ferri-)magnetic phases: some MI , MIIIa, altoghether 31point groups and 275 space groups, but no cubic group.

The antiferromagnetic phases of MnF2, FeF2, CoF2 belong to typeIIIa. For illustration see fig 8 (below).



Figure 8: Simple magnetically ordered structures on a body centered cubic lattice.a) Ferromagnetic structure. b) Antiferromagnetic structure

5.3 Double Space Groups

In section 4 we extended point groups to double groups describing thetransformation of spinors by including an element c0. The elements ofdouble space groups RD can be realized analogously: the rotation partsin the elements {d|t} ∈ R are replaced by the matrices ±D1/2(d), thatis, RD has twice as many elements as R. We write {d±|t} ∈ RD forshort for the two elements related to {d|t} ∈ R. Then RD possesses thedouble translation group TD with the elements {e±|Rh} as an invariantsubgroup whereas TD itself possess the usual translation group T+(∼= T)with {e+|Rh} as an invariant subgroup. Here we define e+ = e ; e− =c0. As TD and T+ are Abelian groups,

TD = T+ × ε , with ε = {{e+|0}, {e−|0}} (52)

is a direct product. Compared to R, RD has the additional elements(d+ = d ; d− = d)

{d−|t} = {e−|Rh} · {d|s(d)} ; t = Rh + s(d) (53)

The group R is not a subgroup of RD, since the elements of R are notclosed with respect to RD.



A Groups and Lie groups

What follows in this appendix is just a short explanation of how someof the groups discussed in this project are connected with Lie groups.

The orthogonal group O(3) is the orthogonal group of n×n matricesfor every dimension n > 0. These matrices form a group because theyare closed under multiplication and when taking inverses. Because theorthogonal group is a set endowed simultaneously with a compatiblestructure of a group and a manifold, it is a Lie group. O(n) has asubmanifold tangentspace at the identity, that is the Lie algebra ofantisymmetric matrices o(n). In fact, the orthogonal group is a compactLie group. In particular O(n, C) is not compact.

The special unitary group S U (n) is the set of n×n unitary matriceswith determinant = +1. S U (2) is homeomorphic with the orthogonalgroup O+

3 . It is also called the unitary unimodular group and is a Liegroup.

The matrices D1/2 that we discussed in section 4 (Double groups),independent of their parametrization, all have an important property;they are unitary matrices with determinant = +1. The set of themobtained by running over the parameter space, build the group S U (2).By construction, each matrix of D1/2 corresponds to a unique rotationof S O(3). S O(3) is the group of all real orthogonal matrices of order3 with determinant = +1. S O(3) describe proper (pure) rotations by afinite angle around any axis. The orthogonal matrices with determinant= −1, corresponding to a rotation followed by an inversion operation,we call the improper rotations.



B Table of crystal systems and Bravais lattices

Number of Schon-System Relations lengths angles Symbol flies

1 Triclinic a 6= b 6= c 6= a 3 3 Γt P Ci

α 6= β 6= γ 6= α

2 Monoclinic a 6= b 6= c 6= a 3 1 Γm P C2h

α = γ = π/2 6= β Cor Γ′

m

α = β = π/2 6= γ A

3 (Ortho-) a 6= b 6= c 6= a 3 - Γo P D2h

rhombic Γ′o C

α = β = γ = π/2 Γ′′o F

Γ′′′o I

4 Tetragonal a = b 6= c 2 - Γq P D4h

(quadratic)α = β = γ = π/2 Γ′

q I

5 Trigonal a = b = c 1 1 Γrh R D3d

(rhombo-hedral) π/2 6= α = β = γ < 2π/3

6 Hexagonal a = b 6= c 2 - Γh P D6h

α = β = π/2;γ < 2π/3 or

H

7 Cubic a = b = c 1 - Γc P Oh

(regular)α = β = γ = π/2 Γ′

c F

Γ′′c I

P: primitive; A: centred on faces C and A, recpectively; F: face.centred (on allfaces); I: body-centred; R: rombohedral; H: Hexagonal.



References

[1] W. Ludwig, C. Falter: Symmetries in Physics, group theory appliedto physical problems, Springer Second edition, (1996)

[2] C. Kittel: Introduction to solid state physics, Wiley and Sons Sev-enth edition, (1996)

[3] http://mathworld.wolfram.com

[4] http://www.solid.phys.ethz.ch/pescia/download/Gt-I-ss03/skriptGt-I-ss03.pdf

[5] http://www1.physik.tu-muenchen.de/lehrstuehle/T30g/lehre/TheoSSP-Skript.pdf


discrete symmetry groups in solid state physics · 2017-02-21 · symmetries, groups and algebras 2...

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