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Applied Group Theory
For Physicists & Chemists
By George H. Duffey Dover Publications, Inc.
Copyright © 1992 George H. Duffey
All rights reserved.
ISBN: 978-0-486-80132-2
CHAPTER 1
Symmetry Operations
1.1
Causality and Symmetry
The universe is not a single indivisible whole. Instead, it consists of parts that can act independently in spite of interactions binding the parts together. This feature allows observers and observing instruments to exist. It also permits both analysis and synthesis to proceed.
An observer first notes that events in his life fall in order; he experiences time locally. Second, whatever he observes can be located at points or small regions in a 3-dimensional space based on his own position at the time of observation. Similar relationships presumably prevail for an observing instrument.
Time is not observed as a global entity but as an independent property of the observing point or small region. It behaves as a directed coordinate orthogonal to the three spatial coordinates of the point. Furthermore, an interval of time can be measured by the distance traveled by a photon in the interval. Consequently, the arena in which phenomena occur is a 4-dimensional continuum in which a displacement may be oriented to be either timelike or spacelike.
In constructing science, one seeks out the patterns that exist among the observations. One presumes that the material world is not capricious or lawless—that it is not governed by spirits as primitive man believed. If certain events appear to follow as a consequence of particular conditions, these events are said to be caused by the conditions. Thus in Newtonian mechanics, one says that the acceleration of a body is caused by the net force acting on it.
In principle, uniqueness need not prevail. A given set of conditions, a given cause, may lead to various possible results rather than to a single result. Then degeneracy is said to obtain. For instance, the radioactive nuclei in a sample may be shown to be identical by statistical tests. Nevertheless, they will disintegrate at random times with a definite half life.
In general, we will call the part of the universe under study a system. The system may be subdivided in various ways. And the resulting parts maybe further subdivided. Each of the subsystems maybe considered a system in its own right in the approximation that it behaves as an entity.
Now, an operation performed on a system may yield an equivalent system with the same spectrum of properties. The entity under study is then said to possess symmetry. The operation is called a symmetry operation.
When these conditions are only approximately satisfied, one says that a near-symmetry exists. The system may then be considered as a pertubation of a corresponding symmetric entity. When the perturbation is small, it may be neglected.
Symmetry operations may act in position space, or in the space-time continuum. They may act in momentum space, or in phase space. Alternatively, they may act in a more general space or plot.
Symmetry operations may also involve other attributes besides position and momentum. Examples of these include particle spin, isotopic spin, hypercharge, color.
1.2
Common Symmetry Operations
For certain properties, the behavior of a system may be governed by a particular function or operator. In classical mechanics, the discriminating function may be a potential, a Lagrangian, or a Hamiltonian. In quantum mechanics, the discriminating operator may be that for some angular momentum or energy.
Now, any system for which distinct operations fail to alter the form of a discriminating function or operator is said to possess symmetry. The operations that leave the pertinent function or operator unchanged are called symmetry operations.
Some of the processes that transform a symmetric region into an equivalent region are geometric, while some are not. Others consist of a geometric change combined with a nongeometric change. Each geometric symmetry operation occurs with respect to a structure in the system, a base.
A nongeometric alteration is called a conversion. In magnetic systems, a conversion involves changing the magnetic state of a particle (as when spins are reversed). In colored systems, a conversion involves changing one color to another in a cycle. In particle systems, a conversion may involve changing one particle into another.
Symmetry operations often met in dealing with the mechanics of macroscopic and microscopic systems are described in Table 1.1. The geometric processes include reorientations about a base, translations, and translations combined with reorientations. In symmetric systems, these appear as permutation of like parts.
1.3
Reorientation Matrices
The reorientations in Table 1.1 may be carried out on the physical system or on the coordinate axes. The first kind is said to be active, the second kind passive. Both kinds are described by homogenous linear transformations of appropriate Cartesian coordinates. Such transformations can be represented by linear matrix equations.
Where there is a point about which the reorientation is executed, this point is chosen as the origin. Where there is more than one such invariant point, a representative point from these points is chosen as the origin. The axes are drawn in appropriate directions.
The coordinates of a typical point of the system before and after the transformation under discussion are designated (x, y, z) and (x', y', z'), respectively. These are related by the equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
which may be abbreviated as
r' = Ar. (1.2)
In the identity operation I there is no change and formula (1.1) reduces to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)
The symbolic form for equation (1.3) is
r' = IR. (1.4)
The base for a 1-dimensional reflection may be made a coordinate plane. When the plane is the yz plane, the operation changes the sign of coordinate x and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)
A symbolic form for equation (1.5) is
r' = σvr. (1.6)
Reflection by a vertical plane is often labeled σv. Reflection by the horizontal xy plane is labeled σh. Operation σd is reflection through a vertical plane that forms part of a dihedral angle of symmetry, or that bisects the angle between successive σv planes.
Reflection through a point is called inversion i. With the origin this point, the matrix equation for the operation is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)
or
r' = ir. (1.8)
Rotation of a physical body counterclockwise by angle φ about the z axis produces the same change in the coordinates of a point in the body as rotation of the coordinate axes clockwise by angle φ about the z axis. From definitions of the sine and cosine, the change is described by the equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.9)
which may be abbreviated as
r' = Cnr (1.10)
if
φ = 2π/n. (1.11)
and
r' = Cnmr (1.12)
when
φ = 2π(m/n). (1.13)
When all symmetry rotations about a given axis have the form Cnm where n is an integer and
m = 1, 2, ..., n - 1, n, (1.14)
then the axis is said to be an n -fold axis. Rotations by 1/2 turn about axes perpendicular to the n-fold axis are labeled C2', C2",.... Operation Cn followed by inversion i through the origin is called rotoinversion. Operation Cn followed by reflection σh with respect to a plane perpendicular to the axis of rotation is called a rotoreflection Sn.
Since a symmetry operation changes a physical system into an equivalent system, it does not introduce any distortion. However, it generally reorients the system. But any matrix that effects a reorientation without distortion is called a reorientation matrix.
Example 1.1
Construct a reorientation matrix for the C3 operation.
Equation (1.9) is the form equation (1.1) assumes when the operation consists of rotation by angle φ about the z axis. When φ is 1/3 turn, the cosine is -1/2 and the sine is √3/2. Then the square matrix in equation (1.9) becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Example 1.2
Construct a matrix that represents reorientation S3.
From the definition, S3 equals rotation C3 followed by reflection σh. When the axis of rotation is the z axis, the reflection changes the sign of z. We have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
1.4 Operations Involving Translations
The reorientations just considered are described by homogenous linear transformations. When the homogeneity is dropped by adding a constant vector to the right side of equation (1.2), the operation includes a translation.
A pure translation entails displacing each point of the system by a constant vector a:
r' = r + a. (1.15)
When the translation is followed by a reorientation effected by matrix R, we have
r" = Rr' = R(r + a) = Rr + Ra = Rr + b. (1.16)
Note that b is the translation Ra.
When a system repeats itself at regular intervals in a certain direction, it is said to possess translational symmetry and to be crystalline in that direction. When it is periodic in three independent directions, the unit that is repeated again and again is called a unit cell One of these cells can be picked as reference and its edges labeled.
a1 = a1e1, a2 = a2e2, a3 = a3e3, (1.17)
where e1, e2, and e3 are the appropriate unit vectors. Then the symmetry translations for the system involve
a = [summation]njaj = [summation]njajej, (1.18)
where n1, n2, and n3 are integers. Equivalent to the point at the origin is the point
r = [summation]njaj. (1.19)
Note how a1, a2, and a3 serve as base vectors.
Designate the angle between a2 and a3 as α, that between a3 and a1 as β, and that between a1 and a2 as γ. Then the unit cells needed to fit observed crystals satisfy the conditions listed in Table 1.2.
If a sinusoidal disturbance
F = F0 sin (k · r - φ) sin ωt (1.20)
is to affect equivalent positions in a crystal in the same manner, wavevector k must be chosen so that k · r increases by an integral number of 2π radians for each symmetry translation a. This condition is satisfied when the wavevector is an integral combination
k = [summation]hjAj (1.21)
of the reciprocal vectors
Aj = 2π aj+1 x aj+2/a1 · a2 x a3. (1.22)
Here h1, h2, and h3 are integers and numbers 1, 2, 3 are considered to be in cyclic order (so 3 + 1 = 1, ...).
For then
Aj · ak = 2πδjk (1.23)
and at equivalent points from equation (1.19) we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.24)
The wavevectors
k = [summation]hjAj, (1.25)
with each hj an integer, define an array of points called the reciprocal lattice. This lattice appears in a plot of the wavevectors, that is, in k-space.
In either k-space or r-space (physical space) the smallest possible unit cell is called a primitive cell. This is not necessarily the same as the conventional unit cell. Thus, the face-centered and the body-centered cubic lattices have rhombohedral primitive cells. Edges a1, a2, a3 of these are illustrated in Figures 1.1 and 1.2.
Example 1.3
A primitive cell of a face-centered cubic lattice is bounded by the vectors
a1 = d/2(i + j), a2 = d/2(j + k), a2 = d/2(i = k),
where d is the length of an edge of the unit cube. What lattice is reciprocal to this lattice?
From the determinant representation of the triple scalar product, we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Then using the determinant representation of the vector product in equation (1.22) leads to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Vectors A1, A2, A3 bound a primitive cell of a body-centered cubic lattice of edge length 4π/d. So the reciprocal lattice for an r-space face-centered cubic lattice is a k-space body-centered cubic lattice.
1.5 Permutation Matrices
Many physical systems are composed of equivalent parts. A symmetry operation then acts by permuting these parts, yielding an equivalent system.
To represent such a process, a person may locate equivalent positions in the equivalent parts and establish a reference point that is not moved during the transformation. The vector drawn from the reference point to the chosen position for the jth part is labeled rj' before, and after, the transformation. One then forms the column matrices
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.26)
in which n is the total number of parts permuted.
From the changes that a given operation causes, one relates the vectors:
rj' = rk. (1.27)
These relations are inserted into matrix r' and matrix r is factored out to give
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.28)
This may be abbreviated as
r' = Ar (1.29)
with A being called a permutation matrix for the operation.
In a cyclic permutation, the transformation does not affect the order of the radius vectors. The matrix A representing such an operation may consist of a sequence of 1's along one diagonal and 0's everywhere else.
A general permutation can be broken down into cyclic permutations. The corresponding matrix partitions into null matrices and matrices representing the cyclic actions.
Example 1.4
Construct a permutation matrix representing the C4 operation.
A system for which C4 is a symmetry operation contains four parts that are permuted cyclicly by the operation. (A simple example appears in Figure 1.3). Initially, the parts are arranged as shown, with vectors r1, r2, r3, r4, r5 drawn from a point on the axis (an invariant point) to the first, second, third, fourth, and fifth parts. After a transformation, the parts have moved, together with their vectors, which are now designated r1', r2', r3', r4', r5'.
Under the C4 operation, the first body moves to the second position, the second body to the third position, the third body to the fourth position, the fourth body to the first position, and the sixth body is merely rotated. Consequently,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
and we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In the last step, the matrix
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Comparing the overall equation with
r' = C4r,
we obtain the permutation matrix
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Example 1.5
Construct a permutation matrix representing a σd operation on the system in Figure 1.3.
Under the σd operation, r1 and r2 are interchanged; also r3 and r4 are interchanged. Thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
With
r' = σdr,
we have the representation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(Continues...)
Excerpted from Applied Group Theory by George H. Duffey. Copyright © 1992 George H. Duffey. Excerpted by permission of Dover Publications, Inc..
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