Three-dimensional close packing in solids is referred to as stacking the second square closed packing exactly above the first. In this tight packing, the spheres are horizontally and vertically correctly balanced. Similarly, we can obtain a simple cubic lattice by adding more layers, one above the other. This can be done in two ways. Three-dimensional close packing from two-dimensional square close-packed layers: By putting the second square closed packing exactly above the first, it is possible to form three-dimensional close packing. In this tight packing, the spheres are horizontally and vertically correctly balanced. Similarly, we can obtain a simple cubic lattice by adding more layers, one above the other.Three-dimensional close packing from two-dimensional hexagonal close-packed layers: With the assistance, of two-dimensional hexagonal packed layers, three-dimensional close packing can be obtained.
What will be the ratio of radii of the spheres in cubic systems simple cubic, body centred cubic and face centred cubic systems. if 'a' stands for the edge length.

Options

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

Explanation

To find the ratio of the radii of the spheres in simple cubic (SCC), body-centered cubic (BCC), and face-centered cubic (FCC) systems, we will derive the relationships between the edge length \( a \) and the radius \( R \) of the spheres for each type of cubic lattice.

### Step 1: Simple Cubic (SCC)
In a simple cubic lattice, the spheres are located at the corners of the cube. The edge length \( a \) is equal to twice the radius \( R \) of the spheres, since the spheres touch each other along the edge.

\[ a = 2R \implies R = \frac{a}{2} \]

### Step 2: Body-Centered Cubic (BCC)
In a body-centered cubic lattice, there is one atom at each corner and one atom at the center of the cube. The relationship between the edge length \( a \) and the radius \( R \) can be derived from the diagonal of the cube. The body diagonal of the cube can be expressed as:

\[ \text{Body diagonal} = \sqrt{3}a \]

This body diagonal consists of 4 radii (since there are two radii from corner spheres and one radius from the center sphere):

\[ \sqrt{3}a = 4R \implies R = \frac{\sqrt{3}a}{4} \]

### Step 3: Face-Centered Cubic (FCC)
In a face-centered cubic lattice, there are atoms at each corner and in the center of each face. The relationship between the edge length \( a \) and the radius \( R \) can be derived from the face diagonal of the cube. The face diagonal can be expressed as:

\[ \text{Face diagonal} = \sqrt{2}a \]

This face diagonal consists of 4 radii (since there are two radii from corner spheres and two from the face-centered spheres):

\[ \sqrt{2}a = 4R \implies R = \frac{\sqrt{2}a}{4} \]

### Step 4: Ratio of Radii
Now we have the expressions for the radius \( R \) in terms of the edge length \( a \):

1. Simple Cubic (SCC): \( R_{SCC} = \frac{a}{2} \)
2. Body-Centered Cubic (BCC): \( R_{BCC} = \frac{\sqrt{3}a}{4} \)
3. Face-Centered Cubic (FCC): \( R_{FCC} = \frac{\sqrt{2}a}{4} \)

To find the ratio of the radii \( R_{SCC} : R_{BCC} : R_{FCC} \):

\[ R_{SCC} : R_{BCC} : R_{FCC} = \frac{a}{2} : \frac{\sqrt{3}a}{4} : \frac{\sqrt{2}a}{4} \]

To simplify this, we can factor out \( a \):

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

Now, to eliminate the fractions, we multiply each term by 4:

\[ = 2 : \sqrt{3} : \sqrt{2} \]

Thus, the final ratio of the radii of the spheres in simple cubic, body-centered cubic, and face-centered cubic systems is:

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