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°.

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:  

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      

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

τ = 1000 Å:          

τ = 750 Å:

τ = 250 Å:

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

Θ = 10°:

Θ = 45°:

B = 0.438°  

Θ = 80°:

Have a nice day!

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .