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1.2: Operator Properties and Mathematical Groups

  • Page ID
    221670
  • The inverse of A (defined as (A)–1) is B if A ⋅ B = E

    For each of the five symmetry operations:

    \(( E )^{-1}= E \Longrightarrow( E )^{-1} \cdot E = E \cdot E = E\)
    \((\sigma)^{-1}=\sigma \Longrightarrow(\sigma)^{-1} \cdot \sigma=\sigma \cdot \sigma= E\)
    \((i)^{-1}=i \Longrightarrow(i)^{-1} \cdot i=i \cdot i=E\)
    \(\left(C_{n}^{m}\right)^{-1}=C_{n}^{n-m} \Longrightarrow\left(C_{n}^{m}\right)^{-1} \cdot C_{n}^{m}=C_{n}^{n-m} \cdot C_{n}^{m}=C_{n}^{n}=E\)
    e.g. \(\left(C_{5}^{2}\right)^{-1}=C_{5}^{3}\) since \(C_{5}^{2} \cdot C_{5}^{3}=E\)
    \(\left(S_{n}^{m}\right)^{-1}=S_{n}^{n-m}(n \text { even }) \Longrightarrow\left(S_{n}^{m}\right)^{-1} \cdot S_{n}^{m}=S_{n}^{n-m} \cdot S_{n}^{m}=S_{n}^{n}=C_{n}^{n} \cdot \sigma_{h}^{n}=E\)
    \(\left(S_{n}^{m}\right)^{-1}=S_{n}^{2 n-m}(n \text { odd }) \Longrightarrow\left(S_{n}^{m}\right)^{-1} \cdot S_{n}^{m}=S_{n}^{2 n-m} \cdot S_{n}^{m}=S_{n}^{2 n}=C_{n}^{2 n} \cdot \sigma_{h}^{2 n}=E\)

    Two operators commute when A ⋅ B = B ⋅ A

    Example: Do C4(z) and σ(xz) commute?

    8.PNG

    … or analyzing with matrix representations,

    \(\left[\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right] \cdot\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{rrr}0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\)

    C4(z) ⋅ σxz = σd´

    Now applying the operations in the inverse order,

    9.PNG

    … or analyzing with matrix representations,

    \(\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{array}\right] \cdot\left[\begin{array}{rrr}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\)

    σxz ⋅ C4(z) = σd

    \begin{equation}
    \therefore \quad C_{4}(z) \sigma(x z)=\sigma_{d}^{\prime} \neq \sigma(x z) C_{4}(z)=\sigma_{d} \Rightarrow \text { so } C_{4}(z) \text { does not commute with } \sigma(x z)
    \end{equation}

    A collection of operations are a mathematical group when the following conditions are met:

    closure: all binary products must be members of the group

    identity: a group must contain the identity operator

    inverse: every operator must have an inverse

    associativity: associative law of multiplication must hold

    (A ⋅ B) ⋅ C = A ⋅ (B ⋅ C

    (note: commutation not required… groups in which all operators do commute are called Abelian)

    Consider the operators C3 and σv. These do not constitute a group because identity criterion is not satisfied. Do E, C3, σv form a group? To address this question, a stereographic projection (featuring critical operators) will be used:

    10.PNG

    So how about closure?

    C3 ⋅ C3 = C3 2 (so C3 2 needs to be included as part of the group)

    11.PNG

    Thus E, C3 and σv are not closed and consequently these operators do not form a group. Is the addition of C3 2 and σv´ sufficient to define a group? In other terms, are there any other operators that are generated by C3 and σv?

    … the proper rotation axis, C3:

    \(C_{3}\)
    \(C _{3} \cdot C _{3}= C _{3}^{2}\)
    \(C _{3} \cdot C _{3} \cdot C _{3}= C _{3}^{2} \cdot C _{3}= C _{3} \cdot C _{3}^{2}= E\)
    \(C _{3} \cdot C _{3} \cdot C _{3} \cdot C _{3}= E \cdot C _{3}= C _{3}\)

    etc.

    \(\therefore C _{3}\) is the generator of \(E , C _{3}\) and \(C _{3}^{2}\), note: these three operators form a group

    … for the plane of reflection, σv

    \(\sigma_{v}\)
    \(\sigma_{v} \cdot \sigma_{v}=E\)
    \(\sigma_{v} \cdot \sigma_{v} \cdot \sigma_{v}=E \cdot \sigma_{v}=\sigma_{v}\)

    etc.
    So we obtain no new information here. But there is more information to be gained upon considering C3 and σv. Have already seen that C3 ⋅ σv = σv’ … how about σv ⋅ C3

    12.PNG

    Will discover that no new operators may be generated. Moreover one finds

    \(\begin{array}{ccccccc} & E ^{-1} & C _{3}^{-1} & \left( C _{3}^{2}\right)^{-1} & \sigma_{ v }^{-1} & \left(\sigma_{ v }^{\prime}\right)^{-1} & \left(\sigma_{ v }^{\prime \prime}\right)^{-1} \\ \text {inverses } & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow & \downarrow \\ & E & C _{3}^{2} & C _{3} & \sigma_{ v } & \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime \prime}\end{array}\)

    The above group is closed, i.e. it contains the identity operator and meets inverse and associativity conditions. Thus the above set of operators constitutes a mathematical group (note that the group is not Abelian).

    Some definitions:

    Operators C3 and σv are called generators for the group since every element of the group can be expressed as a product of these operators (and their inverses).

    The order of the group, designated h, is the number of elements. In the above example, h = 6.

    Groups defined by a single generator are called cyclic groups.

    Example: C3 → E, C3, C3 2

    As mentioned above, E, C3, and C32 meet the conditions of a group; they form a cyclic group. Moreover these three operators are a subgroup of E, C3, C3 2, σv, σv’,σv”. The order of a subgroup must be a divisor of the order of its parent group. (Example hsubgroup = 3, hgroup = 6 … a divisor of 2.)

    A similarity transformation is defined as: v -1 ⋅ A ⋅ ν = B where B is designated the similarity transform of A by x and A and B are conjugates of each other. A complete set of operators that are conjugates to one another is called a class of the group.

    Let’s determine the classes of the group defined by E, C3, C3 2 , σv, σv’,σv”… the analysis is facilitated by the construction of a multiplication table

    \[\begin{array}{l|llllll}
    & E & C _{3} & C _{3}^{2} & \sigma_{ v } & \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime \prime} \\
    \hline E & E & C _{3} & C _{3}^{2} & \sigma_{ v } & \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime \prime} \\
    C _{3} & C _{3} & C _{3}^{2} & E & \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime \prime} & \sigma_{ v } \\
    C _{3}^{2} & C _{3}^{2} & E & C _{3} & \sigma_{ v }^{\prime \prime} & \sigma_{ v } & \sigma_{ v }^{\prime} \\
    \sigma_{ v } & \sigma_{ v } & \sigma_{ v }^{\prime \prime} & \sigma_{ v }^{\prime} & E & C _{3}^{2} & C _{3} \\
    \sigma_{ v }^{\prime} & \sigma_{ v }^{\prime} & \sigma_{ v } & \sigma_{ v }^{\prime \prime} & C _{3} & E & C _{3}^{2} \\
    \sigma_{ v }^{\prime \prime} & \sigma_{ v }^{\prime \prime} & \sigma_{ v }^{\prime} & \sigma_{ v } & C _{3}^{2} & C _{3} & E
    \end{array}\]

    may construct easily using stereographic projections

    \(E ^{-1} \cdot C _{3} \cdot E = E \cdot C _{3} \cdot E = C _{3}\)
    \(C _{3}^{-1} \cdot C _{3} \cdot C _{3}= C _{3}^{2} \cdot C _{3} \cdot C _{3}= E \cdot C _{3}= C _{3}\)
    \(\left( C _{3}^{2}\right)^{-1} \cdot C _{3} \cdot C _{3}^{2}= C _{3} \cdot C _{3} \cdot C _{3}^{2}= C _{3} \cdot E = C _{3}\)
    \(\sigma _{ v }^{-1} \cdot C _{3} \cdot \sigma_{ v }=\sigma_{ v } \cdot C _{3} \cdot \sigma_{ v }=\sigma_{ v } \cdot \sigma_{ v }^{\prime}= C _{3}^{2}\)
    \(\left(\sigma_{ v }^{\prime}\right)^{-1} \cdot C _{3} \cdot \sigma_{ v }^{\prime}=\sigma_{ v }^{\prime} \cdot C _{3} \cdot \sigma_{ v }^{\prime}=\sigma_{ v }^{\prime} \cdot \sigma_{ v }^{\prime \prime}= C _{3}^{2}\)
    \(\left(\sigma_{ v }^{\prime \prime}\right)^{-1} \cdot C _{3} \cdot \sigma_{ v }^{\prime \prime}=\sigma_{ v }^{\prime \prime} \cdot C _{3} \cdot \sigma_{ v }^{\prime \prime}=\sigma_{ v }^{\prime \prime} \cdot \sigma_{ v }= C _{3}^{2}\)

    ∴ C3 and C3 2 from a class

    Performing a similar analysis on σv will reveal that σv, σv’ and σv’’ form a class and E is in a class by itself. Thus there are three classes:

    \(E ,\left( C _{3}, C _{3}^{2}\right),\left(\sigma_{ v }, \sigma_{ v }^{\prime}, \sigma_{ v }^{\prime \prime}\right)\)

    Additional properties of transforms and classes are:

    • no operator occurs in more than one class
    • order of all classes must be integral factors of the group’s order
    • in an Abelian group, each operator is in a class by itself.
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