A frst order reflection from (111) plane is observed for LiX with 2theta =24.6 degree (X-ray of wavelength 1.54 angstrom). Assuming LiX to be a cubic crystal system, calculate the length of the side of the unit cell in angstrom. This is a question of JAM 2014, here, the length is considered as a=2a for ionic system in a solution provided by Arihant's book. Why is it so? Is it correct?

Question

Answer

Length of side of unit cell in a cubic crystal system

To calculate the length of the side of the unit cell in angstrom, we need to consider the information given in the problem statement.

Given:
- Reflection from (111) plane observed for LiX
- 2theta = 24.6 degrees
- X-ray wavelength = 1.54 angstrom

Step 1: Calculate the angle of incidence (theta)
The angle of incidence (theta) can be calculated by dividing the 2theta value by 2. In this case:

theta = 24.6 degrees / 2 = 12.3 degrees

Step 2: Calculate the interplanar spacing (d)
The interplanar spacing (d) can be calculated using Bragg's Law:

nλ = 2d sin(theta)

Where:
- n is the order of reflection (1 for first order)
- λ is the wavelength of X-ray
- theta is the angle of incidence in radians

Converting theta from degrees to radians:

theta (radians) = 12.3 degrees x (π / 180 degrees) ≈ 0.214 radians

Now we can calculate the interplanar spacing (d):

d = λ / (2 sin(theta))

Substituting the given values:

d = 1.54 angstrom / (2 sin(0.214)) ≈ 1.54 angstrom / (2 x 0.211) ≈ 3.65 angstrom

Step 3: Calculate the length of the side of the unit cell (a)
In a cubic crystal system, the interplanar spacing (d) is related to the length of the side of the unit cell (a) as follows:

d = a / √(h^2 + k^2 + l^2)

Where:
- h, k, l are the Miller indices of the plane

For the (111) plane, the Miller indices are (1,1,1). Substituting the values:

3.65 angstrom = a / √(1^2 + 1^2 + 1^2)

Simplifying the equation:

3.65 angstrom = a / √3

Multiplying both sides by √3:

a = 3.65 angstrom x √3 ≈ 6.31 angstrom

Explanation of considering a=2a for ionic system
In ionic systems, the unit cell contains the ions and the distance between the ions is considered as the length of the side of the unit cell (a). However, since the ions are not point-like particles, but have a certain size, it is common practice to consider the effective ionic radius as the distance between the ions.

The effective ionic radius is usually taken as half of the distance between the centers of adjacent ions. Therefore, the length of the side of the unit cell (a) is considered as 2 times the ionic radius (2a).

This assumption helps in simplifying calculations and represents a reasonable approximation for ionic systems. However, it is important to note that this assumption may not hold true for all crystal systems or materials.

In conclusion, for the given problem, the length of the side of the unit cell in angstrom is

Similar IIT JAM Doubts

A first order reflection from (111)plane is observed forLiX with 2 thita=24.6(X ray of wave length 1.54 angstrom). Assuming LiX to be a cubic Crystal system, calculate the length of the side of the unit cell in angstrom?

X-ray diffraction of a cubic crystal gives an intensity maximum for Bragg angle of 20corresponding to the (110) plane. The lattice parameter of the crystal is __________nm.(Consider wavelength of X-ray = 0.15 nm) Correct answer is between '0.28,0.35'. Can you explain this answer?

In a diffraction experiment with X ray of wavelength 1.54 A , diffraction line corresponding to 2theta=20.8 is observed. the interplanner distance is?

We have CFSC of [Co(H2O)6]2 is 7360cm-1 how to calculate delta octahedral by this in kg joule per mole and max wavelength? This is jam 2015 question and 2018 integer type, help me friends?

Top Courses for IIT JAM

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams