Question

Calculate the breadth B ( in degrees of 2θ ), due to the small crystal effect alone, of the powder pattern lines of particles of diameter 1000, 750, and 250Å. Assume θ=45° and λ=1.5 Å. For particles 250 Å in diameter, calculate the breadth B for θ=10, 45, and 80°.

Answer 1

Answer:

For differents τ:  

τ = 1000 Å → B = 0.11°

τ = 750 Å → B = 0.15°

τ = 250 Å → B = 0.44°

For differents Θ:

Θ = 10° → B = 0.31°

Θ = 45° → B = 0.44°

Θ = 80° → B = 1.78°

Explanation:

To factor B is related to the size of particles, Θ, and λ by the Scherrer equation:  

$\tau = \frac{K \lambda}{ B cos(\theta)}$

where τ: size of the particles, λ: is the wavelenght of the X-Rays, B: is the line broadening at half the maximum intensity, Θ: angle of incidence and K: is a shape factor with typical value of 0.9      

$B = \frac{K \lambda}{ \tau cos(\theta)}$

Now, factor B for the diameter of the particles (τ) is:

τ = 1000 Å:          

$B = \frac{0.9 \cdot 1.5}{1000 \cdot cos(45)} = 1.91\cdot 10^{-3} rad = 0.109 ^{\circ}$                

τ = 750 Å:

$B = \frac{0.9 \cdot 1.5}{750 \cdot cos(45)} = 2.54\cdot 10^{-3} rad = 0.146 ^{\circ}$

τ = 250 Å:

$B = \frac{0.9 \cdot 1.5}{250 \cdot cos(45)} = 7.64\cdot 10^{-3} rad = 0.438 ^{\circ}$

For τ = 250 Å, factor B for angles of incidence is:  

Θ = 10°:

$B = \frac{0.9 \cdot 1.5}{250 \cdot cos(10)} = 5.48 \cdot 10^{-3} rad = 0.314 ^{\circ}$

Θ = 45°:

B = 0.438°  

Θ = 80°:

$B = \frac{0.9 \cdot 1.5}{250 \cdot cos(80)} = 0.031 rad = 1.78 ^{\circ}$  

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