Question

For a body centered cubic (bcc) system, the x-ray diffraction peaks are observed for the following \(h^2+k^2+l^2\) value(s)

• A. 3
• B. 4
• C. 5
• D. 7

Hint:

It is very helpful to memorize the selection rules for the common cubic lattices: - \textbf{Simple Cubic (sc):} All (hkl) reflections are allowed. - \textbf{Body-Centered Cubic (bcc):} Allowed if \(h+k+l\) is even. - \textbf{Face-Centered Cubic (fcc):} Allowed if h, k, l are all even or all odd (unmixed). Also, remember that numbers like 7, 15, 23, 28... cannot be expressed as the sum of three squares, so they are forbidden for all cubic lattices.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
X-ray diffraction (XRD) from a crystal lattice produces constructive interference (peaks in the diffraction pattern) only for specific planes, described by Miller indices (hkl). The condition for which planes produce diffraction peaks is called the selection rule, which depends on the crystal structure (e.g., simple cubic, bcc, fcc).
Step 2: Key Formula or Approach:
The selection rule for a body-centered cubic (bcc) lattice is that diffraction peaks are observed only for Miller indices (hkl) where the sum \(h+k+l\) is an even number. We need to check which of the given values of \(S = h^2+k^2+l^2\) can be formed by integers h, k, l that satisfy this condition.
Step 3: Detailed Explanation:
Let's examine the possible values of \(S = h^2+k^2+l^2\) for small integer values of h, k, l and check the bcc selection rule (\(h+k+l = \text{even}\)). - S = 1: Possible (hkl) is (100). Sum \(h+k+l = 1\) (odd). Forbidden.
- S = 2: Possible (hkl) is (110). Sum \(h+k+l = 2\) (even). Allowed.
- S = 3 (Option A): Possible (hkl) is (111). Sum \(h+k+l = 3\) (odd). Forbidden. Thus (A) is incorrect.
- S = 4 (Option B): Possible (hkl) is (200). Sum \(h+k+l = 2\) (even). Allowed. Thus (B) is correct.
- S = 5 (Option C): Possible (hkl) is (210). Sum \(h+k+l = 3\) (odd). Forbidden. Thus (C) is incorrect.
- S = 6: Possible (hkl) is (211). Sum \(h+k+l = 4\) (even). Allowed.
- S = 7 (Option D): 7 cannot be written as the sum of three integer squares. (\(1^2+1^2+1^2=3\), \(2^2+1^2+1^2=6\), \(2^2+2^2+1^2=9\)). So, no plane corresponds to \(h^2+k^2+l^2=7\). It is a "forbidden" value for all cubic lattices. Thus (D) is incorrect. Let's summarize the first few allowed reflections for bcc:

(110): \(h+k+l=2\) (even), \(S=1^2+1^2+0^2=2\)
(200): \(h+k+l=2\) (even), \(S=2^2+0^2+0^2=4\)
(211): \(h+k+l=4\) (even), \(S=2^2+1^2+1^2=6\)
(220): \(h+k+l=4\) (even), \(S=2^2+2^2+0^2=8\)
(310): \(h+k+l=4\) (even), \(S=3^2+1^2+0^2=10\)
(222): \(h+k+l=6\) (even), \(S=2^2+2^2+2^2=12\)
The allowed values of \(h^2+k^2+l^2\) are 2, 4, 6, 8, 10, 12, ... (even numbers, excluding those like 7 that can't be formed). Of the given options, only \(S=4\) corresponds to an allowed reflection for a bcc lattice, which is the (200) plane. Step 4: Final Answer:
For a bcc system, diffraction peaks are observed only when \(h+k+l\) is even. Out of the given options for \(S=h^2+k^2+l^2\), only \(S=4\) can be formed by a set of Miller indices (200) that satisfies the condition \(h+k+l=2+0+0=2\) (even). Therefore, only option (B) is correct.