Crystallography

Step 1: Understanding the Concept:
This is a factual question about the mathematical classification of crystal symmetries. A space group is the complete symmetry group of a crystal, including all translational symmetries (lattice translations, glide planes, screw axes) and point group symmetries (rotations, reflections, inversions). The question asks for the total number of unique space groups that can exist in a three-dimensional periodic structure.
Step 2: Detailed Explanation:
The derivation of the number of space groups is a complex task in mathematical crystallography. It involves combining the 14 fundamental Bravais lattices with the 32 possible crystallographic point groups. The combination of translational symmetry operations (like screw axes and glide planes) with the point group operations leads to a finite and specific number of possible unique space groups.
Through systematic enumeration, it was proven in the late 19th century that there are exactly 230 distinct space groups in three dimensions. This is a fundamental result in the field of crystallography.
Step 3: Final Answer:
There are 230 distinct space groups possible in 3-dimensions.

Step 1: Understanding the Concept:
This question requires the calculation of the interplanar spacing, , for a specific crystal plane in a simple cubic lattice using the given lattice constant.
Step 2: Key Formula or Approach:
For a cubic crystal system (simple, body-centered, or face-centered), the interplanar spacing for a plane with Miller indices (hkl) is given by the formula:

where 'a' is the lattice constant.
Step 3: Detailed Explanation:
We are given:


Lattice constant, m
Miller indices, (hkl) = (034)
First, calculate the sum of the squares of the Miller indices:

Next, find the square root of this sum:

Finally, substitute the values into the formula:

To match the options, we can write this in scientific notation as:

Step 4: Final Answer:
The interplanar spacing for the (034) plane is m.

Step 1: Understanding the Concept:
This question relates to rotational symmetry in crystallography. In a periodic crystal lattice, not all rotational symmetries are possible. The crystallographic restriction theorem dictates which rotational symmetries can exist.
Step 2: Detailed Explanation:
The crystallographic restriction theorem states that if a crystal has rotational symmetry, the order of rotation (n-fold) can only be 1, 2, 3, 4, or 6.


One-fold symmetry (360 rotation): This is trivial and exists in all objects.
Two-fold symmetry (180 rotation): Exists.
Three-fold symmetry (120 rotation): Exists.
Four-fold symmetry (90 rotation): Exists.
Six-fold symmetry (60 rotation): Exists.
A five-fold rotational symmetry (72 rotation) is not allowed because pentagons cannot tile a two-dimensional or three-dimensional space without leaving gaps or overlapping. This would violate the periodic, repeating nature of a crystal lattice.
Step 3: Final Answer:
Therefore, five-fold symmetry does not exist in periodic crystals.

Step 1: Understanding the Concept:
Miller indices are a notation system in crystallography for planes in crystal (Bravais) lattices. The question asks for the correct sequence of steps to determine these indices for a given plane.
Step 2: Detailed Explanation:
The correct procedure for determining Miller indices (hkl) is as follows:
1. Find the intercepts: Determine the points where the plane intersects the crystallographic axes (x, y, z). These intercepts are expressed as multiples of the lattice parameters (e.g., , , ). This corresponds to step D.
2. Take the reciprocal: Take the reciprocals of the numerical parts of the intercepts ( , , ). If a plane is parallel to an axis, its intercept is at infinity ( ), and the reciprocal is zero. This corresponds to step A.
3. Simplify the fraction (Clear fractions): Multiply or divide the reciprocals by a common factor to reduce them to the smallest set of integers (h, k, l). This corresponds to step B.
4. Enclose in parentheses: Write the resulting integers in parentheses (hkl) without any commas. This is the Miller index of the plane. This corresponds to step C.
Step 3: Final Answer:
The correct sequence of steps is D, then A, then B, and finally C.