Platinum crystallize in a face-centred cubic crystal with a unite cell length a. the distance between nearest neighbours is:

Options

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

Answer

To find the distance between nearest neighbors in a face-centered cubic (FCC) crystal structure of platinum, we can follow these steps:

### Step 1: Understanding the FCC Structure
In a face-centered cubic unit cell, there are atoms located at each of the eight corners and at the centers of each of the six faces. The nearest neighbors in this structure are located along the face diagonal.

### Step 2: Identify the Relationship Between Edge Length and Atomic Radius
Let the edge length of the unit cell be denoted as \( a \) and the atomic radius of platinum be denoted as \( r \). In an FCC unit cell, the relationship between the edge length \( a \) and the atomic radius \( r \) can be derived from the geometry of the face diagonal.

### Step 3: Calculate the Face Diagonal
The face diagonal of the cube can be calculated using the Pythagorean theorem. The length of the face diagonal \( d \) can be expressed as:
\[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 4: Relate the Face Diagonal to Atomic Radii
In an FCC structure, the face diagonal is equal to four times the atomic radius (since there are two atomic radii at each end of the diagonal):
\[ d = 4r \]
Thus, we can set the two expressions for the face diagonal equal to each other:
\[ a\sqrt{2} = 4r \]

### Step 5: Solve for the Atomic Radius
From the equation \( a\sqrt{2} = 4r \), we can solve for \( r \):
\[ r = \frac{a\sqrt{2}}{4} = \frac{a}{2\sqrt{2}} \]

### Step 6: Find the Distance Between Nearest Neighbors
The distance between nearest neighbors in an FCC structure is equal to twice the atomic radius:
\[ \text{Distance between nearest neighbors} = 2r \]
Substituting the expression for \( r \):
\[ \text{Distance} = 2 \left(\frac{a}{2\sqrt{2}}\right) = \frac{a}{\sqrt{2}} \]

### Final Answer
Thus, the distance between nearest neighbors in a face-centered cubic unit cell of platinum is:
\[ \frac{a}{\sqrt{2}} \]

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