X-ray beam of wavelength undergoes a first order reflection from a crystal when its angle of incident to a crystal face is 26.5°, and an X-ray beam of wavelength 0.097 nm undergoes a third order reflection when its angle of incidence to that face is 60°. Assuming that the two beams reflect from the same family of planes, calculate (a) the interplanar spacing of the planes and (b) wavelength. (Ans: (a) 0.168 nm, (b) 0.150 nm)



Data Given:

For 1st order reflection:

Angle = θ₁ = 26.5° 
Wavelength  = λ₁ = ?
n₁ = 1

For 3rd order reflection:

Angle = θ₂ = 60° 
Wavelength  = λ₂ = 0.097 nm = 0.097 ｘ 10⁻⁹ m
n₂ = 3


To Find:


(a) Crystal Interplanar spacing = d = ?

(b) Wavelength of 1st order reflection = λ₁ = ?


Solution:


(a) Crystal Interplanar spacing = d = ?


Using Braggs Equation (3rd order reflection)

2 d sin θ₂ = m₂ λ₂

or 

d = `\frac {m₂ λ₂}{2 sin θ₂}`

by putting the corresponding values 

d = `\frac {3 ｘ 0.097 ｘ 10⁻⁹ m}{2 sin 60°}`

d = `\frac {0.291 ｘ 10⁻⁹ m}{2 ｘ 0.866}`


by simplifying

d = 0.168 ｘ 10⁻⁹ m

or

d = 0.168 nm ---------------Ans.(1)


(b) Wavelength of 1st order reflection = λ₁ = ?

Again using the brag's formula (for 1st Order reflection)

2d sin θ₁ = m₁ λ₁

or 

λ₁ = `\frac {2d sin θ₁}{m₁}`


by putting the corresponding values

λ₁ = `\frac {2 ｘ 0.168 ｘ 10⁻⁹ m ｘ sin 26.5°}{1}`

λ₁ = 0.336  ｘ 10⁻⁹ m ｘ 0.446

λ = 0.149856 ｘ 10⁻⁹ m 

or

λ₁ = 0.15 nm -----------------------Ans (2)



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