Lecture Note
Principles of Plane Group Derivation
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University:
Massachusetts Institute of Technology -
Course:
3.60 | Symmetry, Structure, and Tensor Properties of Materials Academic year:
2019
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Views:
179
Pages:
19
Author:
Origbo2ol
3.60 Symmetry, Structure and Tensor Properties of Materials
Subheading:
PRINCIPLES OF PLANE GROUP DERIVATION
Note:
(NOTE: THIS MATERIAL IS NOT TREERTED IN THE TEXT!)
Summary of Principles in Advance (That is, an Abstract):
(a) ADD EACH of THE TWO-DIMENSIONAL POINT GROUPS TO EACH of THE TWO-DIMENSIONAL LATTICE NETS which CAN ACCOMODATE THEM. EACH of THE OPERATIONS of THE POINT GROUP COMBINED WITH EACH of THE INDEPENDENT LATTICE TRANSLATIONS which TERMINATES WITHIN THE UNIT CELL MUST CREATE SOME NEW SYMMETRY OPERATION ELSEWHERE within the Cell. LOCATE THESE USING THE "COMBINATION THEOREMS", EXPRESSIONS IN AN ALGEBRA of OPERATIONS THAT SPECIFY THE NET EFFECT of TWO SEQUENTIAL SYMMETRY TRANSFORMATIONS.
T . T = T
T . σ = σ’
σ . T = σ’
σ . σ = T or R
(b) SYMMETRY PLANES IN A POINT GROUP MUST BE MIRROR PLANES, BUT GLIDES ARE PERMITTED IN A PLANE GROUP AS IT IS TRANSLATIONALLY PERIODIC. REPLACE, THEREFORE, EACH of THE PURE MIRROR PLANES IN THE POINT GROUPS of (a) BY GLIDES, AND LOCATE THE ADDITIONAL SYMMETRY ELEMENTS which ARISE USING THE ABOVE COMBINATION THEOREMS.
(c) SYMMETRY PLANES AND AXES NEED NO LONGER INTERSECT IN A PLANE GROUP. THEREFORE, STARTING WITH THE PLANE GROUPS which CONSIST of ROTATION AXES (Alone) PLACED IN A PLANE NET, ATTEMPT TO INTERLEAVE GLIDE PLANES OR MIRROR PLANES WITH THE AXES IN SUCH A WAY THAT NO NEW AXES ARE CREATED. FOR EACH POSSIBLE COMBINATION FIND THE NEW SYMMETRY ELEMENTS which ARISE USING THE ABOVE COMBINATION THEOREMS AND THE NEW THEOREM
(d) THE TWO-DIMENSIONAL LATTICES (NETS)
Enlice we had shown THAT ROTATION AND REFLECTION REQUIRE 5 TYPES of TWO-DIMENSIONAL LATTICES, EACH with A DISTINCT SORT of SPECIALIZATION which is DICTATED BY SYMMETRY
1-fold or 2-fold axis: Parallelogram Net
4-fold axis: Square Net
3-fold or 6-fold axis: Equilateral (or Hexagonal) Net
Mirror Plane: Either a Primitive or Centered (Diamond) Rectangular Net
(e) THE TWO-DIMENSIONAL POINT GROUPS
Let’s NOW ASK HOW ONE CAN COMBINE, ABOUT A FIXED POINT IN SPACE, MORE THAN ONE of THE TWO-DIMENSION SYMMETRY ELEMENTS which CAN SUBSEQUENTLY BE COMBINED WITH A TWO-DIMENSIONAL LATTICE NET NAMELY, 1,2, 3, 4, 6 AND M. A GROUND RULE THAT WE will IMPOSE IS THAT All MAPPINGS of COORDINATES THAT OCCUR MUST INVOLVE ONLY THE TWO DIMENSIONS of THE SPACE. WE will NOT allow AN OPERATION THAT LIFTS A MOTI OUT INTO A THIRD DIMENSION AND THEN RESTORES IT TO THE TWO-DIMENSIONAL SPACE of OUR PLANE (ANY MORE THAN WE WOULD ENTERTAIN AN OPERATION IN THREE DIMENSIONS THAT PLUCKS Things OUT INTO A FOURTH DIMENSION AND THEN POPS THEM SUBSEQUENTLY BACK INTO THERE!
The Single Symmetry Elements That Meet This Restriction Are:
(a) A Rotation Axis (normal to the plane of the 2-D space) which maps things only within the plane normal to the axis.
(b) A Mirror Plane (normal to the plane of the 2-D space) which maps things left-to-right across the mirror line.
If Rotation Axes and Mirror Lines, Separately, Act as 2-Dimensional Symmetry Elements that leave at least one point in space invariant, (they will clearly do so in combination, provided they intersect at a point (which will be the locus that is left invariant).
If we combine two operations, however, we will (like it or not?) introduce a third operation into the space. We will thus need another "combination theorem" that will, once and for all specify the result.
(a) The Combination of a Rotation Axis with a Reflection Operation about a locus 01 is equivalent to a new reflection operation about a locus 02 from the first.
(b) The Combination of Two Successive Reflections Across Loci that intersect at a point.
Note that the initial pair of operations does not commute: change the order and the result is a reflection across a different location - on the other side of the initial reflection locus.
Let the angle between 01 and 02 be defined as α. The successive reflections produce a third motion of the same handedness. This third motion must be a rotation. The amount of rotation is (α + β) but α = β. Thus the pair of reflection operations does not commute either. Change the order and the result is a -2α rotation.
The new symmetry combinations that result from application of either of these theorems are:
2 mm
4 mm
6 mm
Falling Short - Use Can't Simply Call It P3M Because for the First Time We Have Obtained Two Distinct Plane Groups from One and the Same Point Group! Another Convention is Needed to Distinguish Between These.
To Specify the Rule is to Introduce Another Character into the Point Group Part of the Symbol to Specify the Symmetry which is Along the Diagonal of the Cell (The Long Diagonal; The Short One is Symmetry-Equivalent to the Cell Edge) as well as the Symmetry Along the Cell Edge.
The Order of Symbols is
LATTICE TYPE
SYMMETRY ALONG DIAGONAL
SYMMETRY ALONG CELL EDGE
NEW CELL EDGE
OLD POINT GROUP 3M !!
Happily, this Situation Occurs Only for Point Group 3M !!
Given 7 Equilateral Net
...Axes (6, 3, 3-fold) occupy all of the locations of P3. We will Place the Mirrors in the locations of both P31m and P3m1. The Arrangement of Mirrors and Glides which this Results is Simply That for P31m Superposed Upon P3m1. (We will Jumpolistically Not Try to Sketch this one, and Merry Punt P6mm)
P6mm
...Containing the intricate rearrangement of symmetry planes in this plane group is contained in the unit cell.
5.2 Replacement of Mirror Planes by Glide Planes
...Thus far we have Obtained 13 Plane Groups through direct Addition of Point Groups to the 2-dimensional lattices. This might seem to exhaust all possibilities. But in the process of these derivations, we were brought into headlong confrontation with the Glide Plane, a Symmetry Element which we had not encountered earlier as it exists only in translationally periodic patterns and thus plays no role in the point groups.
In considering the Space Groups, however, one is dealing by definition with patterns... that are translationally periodic. So Glide Planes may be present. One might ask whether additional Plane Groups might be Obtained by Replacing Mirror Planes in the Two-Dimensional Point Groups by Glides (which are, Indeed, a Generalization of Mirror Planes That are Translationally Periodic) where Two Independent Mirror Planes are Present in a Point Group, we might Replace One or Both of these by Glides. The following replacements of m's by g's in Point Group Symbol Reason Thereby Present themselves as possibilities.
2mm → 2gg or 2mg
4mm → 4gg or 4mg
6mm → 6gg or 6mg
Principles of Plane Group Derivation
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