<H2>Assignment Problem Set 4<BR>by Nicki Agron-Olshina</H2>

Assignment Problem Set 4
by Nicki Agron-Olshina

Problem 1

State, with a very brief explanation, which of the following items are (i) due to the packing of atoms/molecules in a crystal, (ii) chosen by the crystallographer, and (iii) neither (i) or (ii).

The Unit Cell, (ii) is chosen by the crystallographer generally to contain the smallest number of lattice points or to display the symmetry of the lattice better, e.g., the unit cell may have certain angles constrained to be 90°indicating the presence of an axis.
The Crystal Lattice, (i) is due to the packing of atoms/molecules in a crystal.
The Origin in Reciprocal Space, (i) is due to the packing of atoms/molecules in a crystal, because the reciprocal-space lattice used to identify diffraction spots has a natural origin corresponding to the straight through x-ray beam intersecting the Ewald sphere.
The Origin in the Crystal (ii) In the real spcace of the crystal, the origin of the lattice is an arbitary point chosen by the crystallographer so that it is convenient to describe the contents of the unit cell.
The Space-Group Symmetry of the Crystal (i) is due to the packing of atoms/molecules in a crystal.

Problem 2

Briefly explain, with two examples (in symbol format), what is meant by the term Laue Class.
The term Laue Class (or Laue Group) refers to the eleven centrosymmetric point groups which describe 3-dimentional point groups.
Examples:

2/m - A twofold rotation axis with a mirror plane perpendicular to it has a point of inversion where the axis and mirror plane intersect.
4/m - A fourfold rotation axis with a mirror plane perpendicular to it has a point of inversion where the axis and mirror plane intersect.

Problem 3

Crystallographic symmetry operations may be grouped into 5 distinct groups, namely, lattice translations, rotations, rotary-inversitons, screw-rotations, and mirror/glide planes. Which of these symmetry operations cannot be found in the crystal structure of an enantiomorphic molecule, e.g. a naturally-occuring sugar such as D-glucose? Give brief reasons for your answer.
An enantiomorphic crystal can have no improper rotation axis or symmetry element that can change the handedness of an object within it, thus rotary-inversion axes and mirror/glide planes will not be found in enantiomorphic molecules.

Problem 4

An organometallic cobalt compound crystallises in space group P2₁/c.
If the coordinates of the central metal atom, Co, in this compound are 0.271,0.250,0.458, determine the coordinates of any other Co atoms within the unit cell, showing your workings.
Is the Co atom sited on a special position in this space group? Give brief reasons for your answer.

The following table shows the atomic coordinates for an atom in a general position in the space group P2₁/c. The atomic coordinates were calculated using the symmetry operators for an atom in a general site then, where appropriate, adjusted so that all the atoms would be within the unit cell.

Symmetry Operators Atomic Coordinates

x y z

x, y, z 0.271 0.250 0.458

-x, 1/2+y, 1/2-z 0.729 0.750 0.042

-x, -y, -z 0.729 0.750 0.542

x, 1/2-y, 1/2+z 0.271 0.250 0.958

Symmetry Operators	Atomic Coordinates
	x	y	z
x, y, z	0.271	0.250	0.458
-x, 1/2+y, 1/2-z	0.729	0.750	0.042
-x, -y, -z	0.729	0.750	0.542
x, 1/2-y, 1/2+z	0.271	0.250	0.958

The Cobalt atom is not sited on a special position because it is not located on an axis of rotation or on a mirror plane.

Problem 5

Given the three different views of 1,2-dichloroethane in its most-likely (minimum-energy) conformation, what symmetry elements are present (if any)? Suggest the molecular point symmetry for this particular conformation.
The 1,2 dichloroethane in its most-likely (minimum-energy) conformation contains the following symmetry elements.

A point of inversion located in the middle of the carbon-carbon bond.
A mirror plane which bisects both the chlorine and the carbon atoms and bisects the angle between the two hydrogen atoms attached to each carbon.
A twofold rotation axis perpendicular to the mirror plane and intersecting the mirror plane at the point midway between the two carbon atoms.

The molecular point symmetry for this particular conformation of 1,2-dichloroethane is 2/m.

Problem 6

Given the diagram of the primitive space group shown in Assignment 4:
i. Identify the three types of symmetry element present in the diagram.

Twofold rotation axes parallel to the c-axis.
Mirror planes parallel to both the b- and c-axes that intersect the a-axis at (1/4,0,0,) and (3/4,0,0)
Glide plane perpendicular to the b-axis and the plane of the screen with a glide direction parallel to the a-axis.

ii. Given the position of the right-handed object shown in red on the image, specify (using the letters A-Z, A-i) the position of symmetry-related objects within the unit cell, stating for each the symmetry element that relates the red and magenta objects.

iii. For each symmetry-related object specified in (ii) above, state its handedness giving a brief reason for your answer.

The answers to the two above questions are summarised in the following table.

Objects Position in
Unit Cell Handedness Reason for handedness

Red x, y, z Right Given

H x, 1-y, z Left Lattice translation of (0,1,0) of left-handed object, e, parallel to b-axis.

I 1/2-x, 1-y, z Right Reflection of left-handed object, H, across mirror plane at (1/4,y,z).

J 1/2+x, 1-y, z Left Lattice translation of (0,1,0) of left-handed object, g, parallel to b-axis.

K 1-x, 1-y, z Right Related to right-handed red object by twofold rotation axis parallel to the c-axes and located at (1/2,1/2,z)

N x, 1/2+y, z Right Lattice translation of (0,1/2,0) of right-handed RED object parallel to b-axis.

O 1/2-x, 1/2+y, z Left Reflection of right-handed object, N, across mirror plane at (1/4,y,z).

P 1/2+x, 1/2+y, z Right Lattice translation of right-handed object, N, along a-glide plane.

Q 1-x, 1/2+y, z Left Related to right-handed object, P, by reflection across mirror plane at (3/4,y,z).

T x, 1/2-y, z Left Lattice translation of (0,1/2,0) of left-handed object, e, parallel to b-axis.

U 1/2-x, 1/2-y, z Right Reflection of left-handed object, T, across mirror plane at (1/4,y,z).

V 1/2+x, 1/2-y, z Left Rotation of left-handed object, O, about a twofold rotation axis parallel to the c-axis and located at (1/2, 1/2, z).

W 1-x, 1/2-y, z Right Reflection of left-handed object, V, across mirror plane at (3/4,y,z).

Z 1/2-x, y, z Left Reflection of right-handed RED object, across mirror plane at (1/4,y,z)

a 1/2+x, y, z Right Lattice translation of right-handed RED object along a-glide plane.

b 1-x, y, z Left Reflection of right-handed object, a, across mirror plane at (3/4,y,z).

Objects	Position in Unit Cell	Handedness	Reason for handedness
Red	x, y, z	Right	Given
H	x, 1-y, z	Left	Lattice translation of (0,1,0) of left-handed object, e, parallel to b-axis.
I	1/2-x, 1-y, z	Right	Reflection of left-handed object, H, across mirror plane at (1/4,y,z).
J	1/2+x, 1-y, z	Left	Lattice translation of (0,1,0) of left-handed object, g, parallel to b-axis.
K	1-x, 1-y, z	Right	Related to right-handed red object by twofold rotation axis parallel to the c-axes and located at (1/2,1/2,z)
N	x, 1/2+y, z	Right	Lattice translation of (0,1/2,0) of right-handed RED object parallel to b-axis.
O	1/2-x, 1/2+y, z	Left	Reflection of right-handed object, N, across mirror plane at (1/4,y,z).
P	1/2+x, 1/2+y, z	Right	Lattice translation of right-handed object, N, along a-glide plane.
Q	1-x, 1/2+y, z	Left	Related to right-handed object, P, by reflection across mirror plane at (3/4,y,z).
T	x, 1/2-y, z	Left	Lattice translation of (0,1/2,0) of left-handed object, e, parallel to b-axis.
U	1/2-x, 1/2-y, z	Right	Reflection of left-handed object, T, across mirror plane at (1/4,y,z).
V	1/2+x, 1/2-y, z	Left	Rotation of left-handed object, O, about a twofold rotation axis parallel to the c-axis and located at (1/2, 1/2, z).
W	1-x, 1/2-y, z	Right	Reflection of left-handed object, V, across mirror plane at (3/4,y,z).
Z	1/2-x, y, z	Left	Reflection of right-handed RED object, across mirror plane at (1/4,y,z)
a	1/2+x, y, z	Right	Lattice translation of right-handed RED object along a-glide plane.
b	1-x, y, z	Left	Reflection of right-handed object, a, across mirror plane at (3/4,y,z).

iv. Specify the crystal system to which this space group belongs giving a brief reason for your answer.

This space group belongs to the Orthorhombic mm2 crystal system because it contains a twofold rotation axis parallel to the c-axis in addition to two mirror/glide planes that are perpendicular to each other and parallel to the c-axis.