Question

Spheres of unit diameter are centered at \( (l, m, n) \), where \( l, m, \) and \( n \) take every possible integer value. The distance between two spheres is computed from the center of one sphere to the center of another sphere. For a given sphere, \( x \) is the distance to its nearest sphere and \( y \) is the distance to its next nearest sphere. The value of \( \frac{y}{x} \) is:

• A. \( 2\sqrt{2} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( \sqrt{2} \)
• D. \( 2 \)

Hint:

In lattice problems, the nearest neighbor lies along one axis, and the next nearest is typically along the diagonal. Use the Euclidean distance formula to compute spacing in 3D grids.

Answer 1

The Correct Option is C

Since the spheres are placed on all integer coordinates \((l, m, n)\), they form a cubic lattice.
The distance between any two sphere centers is simply the Euclidean distance between their coordinates.

For a given sphere at \( (0, 0, 0) \), the nearest neighbors are located at a unit distance along the axes:
\[ x = \text{distance to nearest sphere} = \sqrt{1^2 + 0^2 + 0^2} = 1 \]
The next nearest neighbors lie diagonally in the 2D planes, such as \( (1, 1, 0) \), \( (0, 1, 1) \), etc., and their distance is:
\[ y = \sqrt{1^2 + 1^2 + 0^2} = \sqrt{2} \]
Thus, the required ratio is:
\[ \frac{y}{x} = \frac{\sqrt{2}}{1} = \sqrt{2} \]