A 90 KeV X-ray photon is fired at a carbon target and Compton scattering occurs. Find the wavelength of the incident photon and the wavelength of the scattered photon for a scattering angle of (a) 30 (b) 60. (Answer: 13.8 pm  (a) 14.1 pm (b) 15 pm)



Given:

Energy of the incident X-ray photon = E=  90 keV = 90 ｘ 10³  eV = 90 ｘ 10³ ｘ 1.6 ｘ 10⁻¹⁹ J = 144 ｘ 10⁻¹⁶ J

(a) Scattering angle = ፀ₁ = 30°

(a) Scattering angle = ፀ₂ = 60° 

∴ Plank's Constant = h =  6.626 ｘ 10⁻³⁴ m² kg s⁻¹

∴ Rest mass of the electron = m₀ = 9.11 ｘ 10⁻³¹ kg

∴ Speed of light = 3 ｘ 10⁸ m s⁻¹


To Find:

The wavelength of the incident photon = λ =  ?

(a) Wavelength of the scattered photon = λ' =  ?

(b) Wavelength of the scattered photon = λ'' =  ?


Solution:

Using the following formula for finding the wavelength of the incident photon as 

E = `\frac {hc}{λ}`

of

 λ = `\frac {hc}{E}`

By putting values

 λ = `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹ ｘ 3 ｘ 10⁸ m s⁻¹ }{144 ｘ 10⁻¹⁶ J}`

λ = `\frac {19.878 ｘ 10⁻²⁶ m³ kg s⁻² }{144 ｘ 10⁻¹⁶ J }`

λ = 0.1380ｘ 10⁻¹⁰ m

or

λ = 13.8 ｘ 10⁻¹² m

or

λ = 13.8 pm -------------------Ans. 1


(a) Wavelength of the scattered photon = λ' =  ?

Since we have

Δλ = λ' -  λ =  `\frac {h}{m₀c}` (1 - cosፀ)

or 

λ' =  λ   +  `\frac {h}{m₀c}` (1 - cosፀ)

by putting values


λ' =  13.8 pm + `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{9.11 ｘ 10⁻³¹ kg ｘ 3 ｘ 10⁸ m s⁻¹ }` (1 - cos 30°)


λ'  =    13.8 ｘ 10⁻¹² m + `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{27.3 ｘ 10⁻²³ kg m s⁻¹ }` (1 - 0.866) 

λ'  =    13.8 ｘ 10⁻¹² m + 0.2427  ｘ 10⁻¹¹ m ｘ 0.134

λ'  =   13.8 ｘ 10⁻¹² m  + 0.0325218  ｘ 10⁻¹¹ m 

λ'  =   13.8 ｘ 10⁻¹² m  + 0.325  ｘ 10⁻¹² m

λ'  =   14.125 ｘ 10⁻¹² m  

or

λ'  =   14.125 pm  ----------------Ans. 2



(b) Wavelength of the scattered photon = λ'' =  ?



λ'' =  λ   +  `\frac {h}{m₀c}` (1 - cosፀ)

by putting values


λ'' =  13.8 pm + `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{9.11 ｘ 10⁻³¹ kg ｘ 3 ｘ 10⁸ m s⁻¹ }` (1 - cos 60°)


λ''  =    13.8 ｘ 10⁻¹² m + `\frac {6.626 ｘ 10⁻³⁴ m² kg s⁻¹}{27.3 ｘ 10⁻²³ kg m s⁻¹ }` (1 - 0.5) 

λ''  =    13.8 ｘ 10⁻¹² m + 0.2427  ｘ 10⁻¹¹ m ｘ 0.5

λ''  =   13.8 ｘ 10⁻¹² m  + 0.12135  ｘ 10⁻¹¹ m 

λ''  =   13.8 ｘ 10⁻¹² m  + 1.2135  ｘ 10⁻¹² m

λ''  =   15.0135 ｘ 10⁻¹² m  

or

λ''  =   15 pm  ----------------Ans. 3




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