A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is 'a' the closest approach between two atoms in metallic crystal will be

Options

A. `Delta H = Delta U = 0`
B. `|DeltaH| lt |DeltaU|`
C. `|DeltaH| gt |DeltaU|`
D. `Delta H = Delta U ne 0`

Answer

To find the closest approach between two atoms in a face-centered cubic (FCC) structure, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic structure, atoms are located at each corner of the cube and at the center of each face. The edge length of the unit cell is denoted as 'a'.

### Step 2: Identify the Atoms in Contact
In an FCC structure, the closest approach occurs between an atom at the face center and an atom at a corner of the cube. These two atoms are the ones that touch each other.

### Step 3: Determine the Length of the Face Diagonal
To find the closest approach, we need to calculate the length of the face diagonal of the cube. The face diagonal can be calculated using the Pythagorean theorem.

For a cube with edge length 'a':
- The face diagonal (d) can be calculated as:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 4: Calculate the Closest Approach
The closest approach between the two atoms (the distance between them) is half of the face diagonal, since the two atoms are in contact at this point.

Thus, the closest approach (C) is given by:
\[ C = \frac{d}{2} = \frac{a\sqrt{2}}{2} = \frac{a}{\sqrt{2}} \]

### Final Answer
The closest approach between two atoms in a face-centered cubic structure is:
\[ \frac{a}{\sqrt{2}} \]

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