Question

Which one of the following symmetry elements is an INCORRECT representation of rotoinversion operation?

• A. 1A₃+ inversion centre = 3̅
• B. 1A₂ = 4̅
• C. Mirror plane = 2̅
• D. 1A₃/m = 6̅

Answer 1

The Correct Option is B

The given question concerns the identification of incorrect representations of rotoinversion symmetry operations. Let's understand the concept and examine the options one-by-one to determine which one is incorrect.

Understanding Rotoinversion Operations: 

A rotoinversion operation combines both rotation about an axis and inversion through a point. Here are relevant rotoinversion operations:

• 1A_n + inversion centre = n̅: A rotation of \(\frac{360^\circ}{n}\) followed by an inversion.
• 1A_n/m = 2n̅: A mirror reflection and a rotation combine for specific cases (6̅ denotes 1A₃/m).

Evaluating the Options:

1. 1A₃ + inversion centre = 3̅: This is correct as it involves a 120-degree (3-fold) rotation followed by an inversion.
2. 1A₂ = 4̅: Incorrect. The notation 4̅ indicates a four-fold rotoinversion. However, 1A₂ implies a two-fold axis which should normally combine with inversion to give 2̅, not 4̅. Thus, this is an incorrect representation.
3. Mirror plane = 2̅: Incorrect, but not selected as an option, as 2̅ implies a two-fold inversion, not just a mirror plane.
4. 1A₃/m = 6̅: Correct. A combination of a three-fold rotation and a mirror plane(left-handed) alignment denotes the 6̅ operation.

Conclusion:

The incorrect representation is 1A₂ = 4̅, because it incorrectly uses four-fold notation for what should understand through two-fold symmetry paired with inversion.