If `'a'` stands for the edge length of the cubic systems: simple cubic, body centred cubic and face centred cubic then the ratio of radii of the spheres in these systems will be respectively,

Options

A. `(1)/(2)a:(sqrt3)/(2)a:(sqrt2)/(2)a`
B. `1a: sqrt3a:sqrt2a`
C. `(1)/(2)a:(sqrt3)/(4)a:(1)/(2sqrt2)a`
D. `(1)/(2)a:sqrt3a:(1)/(sqrt2)a`

Answer

To find the ratio of the radii of the spheres in simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) systems, we will derive the radius of the spheres in each cubic system based on the edge length 'a'.

### Step 1: Simple Cubic (SC)
In a simple cubic unit cell, atoms are located at each corner of the cube. The relationship between the edge length 'a' and the radius 'r' of the atom can be expressed as follows:

- The edge length 'a' is equal to twice the radius of the atom:
\[ a = 2r_{SC} \]
- Therefore, the radius of the atom in a simple cubic structure is:
\[ r_{SC} = \frac{a}{2} \]

### Step 2: Body-Centered Cubic (BCC)
In a body-centered cubic unit cell, there is one atom at each corner and one atom at the center of the cube. To find the radius, we can use the diagonal of the cube:

- The body diagonal of the cube can be calculated using the Pythagorean theorem:
\[ \text{Body diagonal} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3}a \]
- The body diagonal is also equal to four times the radius of the atom:
\[ \text{Body diagonal} = 4r_{BCC} \]
- Equating the two expressions gives:
\[ 4r_{BCC} = \sqrt{3}a \]
- Thus, the radius of the atom in a body-centered cubic structure is:
\[ r_{BCC} = \frac{\sqrt{3}}{4}a \]

### Step 3: Face-Centered Cubic (FCC)
In a face-centered cubic unit cell, there are atoms at each corner and one atom at the center of each face. To find the radius, we can analyze the face diagonal:

- The face diagonal can be calculated as:
\[ \text{Face diagonal} = \sqrt{a^2 + a^2} = \sqrt{2}a \]
- The face diagonal is also equal to four times the radius of the atom:
\[ \text{Face diagonal} = 4r_{FCC} \]
- Equating the two expressions gives:
\[ 4r_{FCC} = \sqrt{2}a \]
- Thus, the radius of the atom in a face-centered cubic structure is:
\[ r_{FCC} = \frac{\sqrt{2}}{4}a = \frac{a}{2\sqrt{2}} \]

### Step 4: Ratio of Radii
Now we can find the ratio of the radii of the spheres in the three cubic systems:

\[ \text{Ratio} = r_{SC} : r_{BCC} : r_{FCC} = \frac{a}{2} : \frac{\sqrt{3}}{4}a : \frac{a}{2\sqrt{2}} \]

To simplify, we can factor out 'a' from each term:

\[ \text{Ratio} = \frac{1}{2} : \frac{\sqrt{3}}{4} : \frac{1}{2\sqrt{2}} \]

To express this ratio without fractions, we can multiply each term by 4:

\[ \text{Ratio} = 2 : \sqrt{3} : \frac{2}{\sqrt{2}} = 2 : \sqrt{3} : \sqrt{2} \]

### Final Answer
The ratio of the radii of the spheres in simple cubic, body-centered cubic, and face-centered cubic systems is:
\[ 2 : \sqrt{3} : \sqrt{2} \]