Question

Axis of symmetry in crystals is given by the general formula (a) \(\frac{360^{\circ}}{n}\) (b) \(n \times 360^{\circ}\) (c) \(n \times 180^{\circ}\) (d) \(n / 90^{\circ}\)

Short answer

None of the given options is a correct formula for the axis of symmetry in crystals. A probable answer might be \(\frac{360^{\circ}}{n}\), but this isn't among the given options.

Step-by-step solution

Understanding Symmetry in Crystals

In crystallography, the axis of symmetry is termed as such because it tells us how many times a pattern repeats around a central point. The information provided allows us to discern that 'n' in our problem refers to the number of times the pattern repeats.

Analysis of given options

The axis of symmetry in crystals is given by the formula which, when multiplied by the symmetry (n), will yield a full rotation (360 degrees). We analyze the provided options: (a) \(\frac{360^{\circ}}{n}\) leads to a decreasing angle with increasing symmetry, which is not logical. (b) \(n \times 360^{\circ}\) leads to a result greater than 360 degrees which does not fit within a circle's confines. (c) \(n \times 180^{\circ}\) gives a value exceeding 180 degrees for symmetry more than 1 which is beyond the highest possible symmetry on a plane (180 degrees). (d) \(n / 90^{\circ}\) isn't coherent because n (a counting number or integer) divided by an angle would not generate an interpretable output.

Conclusion of Correct Option

Thus, analytically none of the options seem to fit the requirement of a formula for the axis of symmetry in crystals. The usual convention in such cases might be to consider the reciprocal relationship. That might lead to a formula equivalent to option (a), i.e., \( \frac{360^{\circ}}{n} \). It is suggested to review the problem because currently none of the alternatives seem to provide a satisfactory answer.

Key concepts

Understanding Crystallography

Crystallography is the study of the arrangement of atoms in the solid state, particularly in crystalline solids. Within these structures, atoms are organized in a highly ordered pattern that extends in all three dimensions. This pattern is known as a crystal lattice and is a defining characteristic of a crystal. Each point within the lattice represents the position of an atom, a molecule, or a group of atoms.

One of the fundamental aspects of crystallography is the identification of the unit cell, the simplest repeating unit in the lattice that, when stacked together with identical cells, forms the entire crystal. By understanding the geometric shape and dimensions of the unit cell, one can infer the symmetry and physical properties of the crystal. Symmetry, which is a measure of the repetition of the pattern in different directions, plays an integral role in determining how light interacts with the crystal, its mechanical strengths, and other physical properties.

Symmetry Operations in Crystals

Symmetry operations in crystallography are the movements that can be applied to a crystal whereby, after the movement has been undertaken, the crystal appears unchanged. These operations include rotation, reflection, inversion, and translation. An understanding of symmetry operations is crucial because it helps us to classify crystals into symmetry groups and to understand their physical properties and behavior.

These operations are applied with respect to symmetry elements, which include points, lines, and planes. An axis of symmetry is one such line-based element; it's an imaginary line where if the crystal is rotated a certain angle about this axis, the crystal appears the same as before the rotation. This concept is essential for categorizing crystals and has a direct link to the physical properties of the crystal such as optical activity, cleavage planes, and more.

Rotational Symmetry in Crystals

Rotational symmetry in crystals is a type of symmetry operation related to an axis of symmetry. If a crystal exhibits rotational symmetry, it means that the crystal can be rotated by a certain angle around the axis and the crystal structure or pattern repeats itself. This angle is specifically the division of 360 degrees by a symmetry number, denoted as 'n', which represents the number of times the pattern repeats during a full 360 degree turn.

The formula \( \frac{360^\circ}{n} \) therefore describes the angle of rotation that preserves the symmetry of the crystal. For instance, a four-fold rotational symmetry means the crystal can be turned 90 degrees about the axis and still look the same, indicating \('n' = 4\), each 90 degrees completes one quarter of a full rotation, thereby obeying the formula. Uncovering the rotational symmetry provides insights into the crystal structure which are vital for understanding material properties, such as how it interacts with electromagnetic waves or mechanical forces.