The ratio of inner planar distances of three types of planes `(d_100,d_110,d_111)` for simple cubic lattice are

Options

A. `1: (1)/(sqrt2) : (1)/(sqrt3)`
B. `(1)/(sqrt3) : (1)/(sqrt2) : 1`
C. `sqrt3 : sqrt2 : 1`
D. `1 : sqrt2 : sqrt3 `

Explanation

To find the ratio of inner planar distances \( (d_{100}, d_{110}, d_{111}) \) for a simple cubic lattice, we will use the formula for interplanar distance, which is given by:

\[ d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \]

where \( a \) is the edge length of the cubic unit cell, and \( (h, k, l) \) are the Miller indices of the plane.

### Step 1: Calculate \( d_{100} \)

For the \( (100) \) plane:
- Miller indices: \( h = 1, k = 0, l = 0 \)
- Using the formula:

\[ d_{100} = \frac{a}{\sqrt{1^2 + 0^2 + 0^2}} = \frac{a}{\sqrt{1}} = a \]

### Step 2: Calculate \( d_{110} \)

For the \( (110) \) plane:
- Miller indices: \( h = 1, k = 1, l = 0 \)
- Using the formula:

\[ d_{110} = \frac{a}{\sqrt{1^2 + 1^2 + 0^2}} = \frac{a}{\sqrt{2}} \]

### Step 3: Calculate \( d_{111} \)

For the \( (111) \) plane:
- Miller indices: \( h = 1, k = 1, l = 1 \)
- Using the formula:

\[ d_{111} = \frac{a}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{a}{\sqrt{3}} \]

### Step 4: Form the ratio \( d_{100} : d_{110} : d_{111} \)

Now, we can express the ratios of the distances:

\[ d_{100} : d_{110} : d_{111} = a : \frac{a}{\sqrt{2}} : \frac{a}{\sqrt{3}} \]

To eliminate \( a \) from the ratio, we can write:

\[ 1 : \frac{1}{\sqrt{2}} : \frac{1}{\sqrt{3}} \]

### Step 5: Simplify the ratio

To express this ratio in a more standard form, we can multiply through by \( \sqrt{6} \) (the least common multiple of the denominators):

\[ \sqrt{6} : \sqrt{6} \cdot \frac{1}{\sqrt{2}} : \sqrt{6} \cdot \frac{1}{\sqrt{3}} = \sqrt{6} : \sqrt{3} : \sqrt{2} \]

Thus, the final ratio of the inner planar distances is:

\[ 1 : \frac{1}{\sqrt{2}} : \frac{1}{\sqrt{3}} \]

### Final Answer

The ratio of inner planar distances \( (d_{100}, d_{110}, d_{111}) \) for a simple cubic lattice is:

\[ 1 : \frac{1}{\sqrt{2}} : \frac{1}{\sqrt{3}} \]