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Unit I: Quantum Physics Lectures on Engineering Physics BE - 201 1.3 Compton Effect A. H. Compton observed that “when a monochromatic beam of high frequency (lower wavelength) radiation (e.g., X-rays and γ-ray) is scattered by a substance, the scattered radiation contains two type of wavelengths one having same wavelength as that of incident radiation while the other having the wavelength greater (or lower frequency) than that of incident radiations. This effect is known as Compton Effect. Quantum Explanation: The explanation was given by Compton which was based on quantum theory of light. According to quantum theory when photon of energy hυ strikes with the substance some of the energy of photon is transferred to the electrons, therefore the energy (or frequency) of photon reduces and wavelength increases. Various assumptions were made for explaining the effect these were: (i) (ii) (iii) Compton Effect is the result of interaction of an individual particle and free electron of target. The collision is relativistic and elastic. The laws of conservation of energy and momentum hold good. Figure 1.6 : Compton Scattering 1.3.1 Derivation of Compton Shift Before collision The energy of incident photon The energy of electron The momentum of the photon The momentum of electron h mo c 2 h c =0 After collision h mc 2 h c = mv Dr. Amita Maurya,, Asstt. Prof., Physics, People’s College of Research & Technology, Bhopal For more lecture notes visit http://sites.google.com/site/dramitamaurya 1 Unit I: Quantum Physics Lectures on Engineering Physics BE - 201 The energy of the system before collision = h mo c 2 The energy of the system after collision = h mc 2 According to the principle of conservation of energy h mo c 2 h mc 2 i.e. mc 2 h h mo c 2 …….. (1.32) According to the principle of conservation of linear momentum along and perpendicular to the direction of incident photon (i.e., along x and y axis), we have h h 0 cos mv cos c c i.e. mvc cos h h cos And 0 i.e. mvc sin h sin …….. (1.33) h sin mv sin c ……..(1.34) Squaring (1.33) and (1.34) and then adding, we get m 2 v 2 c 2 h h cos h sin 2 2 Or m2v 2c 2 h h cos 2 2 h h cos h sin 2 Or m2v 2c 2 h h 2 h h cos 2 2 2 2 2 ……..(1.35) Squaring equation (1.32), we get m2c 4 mo 2c 4 h h 2 h h 2mo c 2 h h 2 2 ……..(1.36) Subtracting (1.35) from (1.36), we get m2c4 m2v2c2 mo 2c4 2 h h cos 1 2moc2 h h ……..(1.37) According to the theory of relativity Dr. Amita Maurya,, Asstt. Prof., Physics, People’s College of Research & Technology, Bhopal For more lecture notes visit http://sites.google.com/site/dramitamaurya 2 Unit I: Quantum Physics m mo 1 v2 c2 , m2 Lectures on Engineering Physics BE - 201 mo 2 v2 1 2 c v2 m2 1 2 mo 2 c Or m 2 c 2 m 2 v 2 mo 2 c 2 Multiplying both sides by c2, we get m2 c 4 m2 v 2 c 2 mo 2c 4 ……..(1.38) Using equation (1.38) equation (1.37) becomes 0 2 h h cos 1 2moc2 h h 2 h h 1 cos 2moc2 h h Or h 1 cos mo c 2 1 1 h 1 cos mo c 2 ……..(1.39) To find the relation in term of wavelength, let us substitute c and c , we thus have Compton shift h 1 cos mo c h 1 cos mo c ……..(1.40) From above equations (1.39) and (1.40) following conclusions can be drawn 1. 2. 3. 4. 5. The wavelength of the scattered photon λ’ is greater than the wavelength of incident photon λ. ∆ λ is independent of the incident wavelength. ∆ λ have the same value for all substance containing free electron ∆ λ only depend on the scattering angle . when = 0; cos = 1 Dr. Amita Maurya,, Asstt. Prof., Physics, People’s College of Research & Technology, Bhopal For more lecture notes visit http://sites.google.com/site/dramitamaurya 3 Unit I: Quantum Physics Lectures on Engineering Physics BE - 201 ∆ λ = λ’ – λ = 0 6. = λ’ = λ, the scattered wavelength is same as the incident wavelength in the direction of incidence. when = ; cos =0 ∆ λ = λ’ – λ = h mo c h 0.02426 Ao = λc ; mo c ……..(1.41) Where λc is called the Compton wavelength of the electron. 7. when = ; cos = -1 max 2h 0.04652 Ao mo c ……..(1.42) Why Compton Effect is not observed in visible spectrum The maximum change in wavelength max is 0.04652 Ao or roughly 0.05 Ao. This small therefore can not be observed for wavelength longer than few angstrom units. For exampleIncident Radiation X-ray Visible Incident max Percentage of incident radiation Wavelength 1 Ao 0.05 Ao 5% (detectable) o 5000 A o 0.05 A 0.001% (undetectable) 1.3.2 Direction of Recoil electron Dividing equation (1.36) by (1.35) direction of recoil electron is given by tan h sin h h cos tan c sin c c cos tan sin cos ……..(1.43) Dr. Amita Maurya,, Asstt. Prof., Physics, People’s College of Research & Technology, Bhopal For more lecture notes visit http://sites.google.com/site/dramitamaurya 4 Unit I: Quantum Physics Lectures on Engineering Physics BE - 201 1.3.3 Kinetic Energy of Recoil Electron The kinetic energy gained by electron is equal to the energy loss by the scattered photon m mo c2 h h m mo c 2 hc E hc Where h 1 cos mo c ……..(1.44) 1.3.4 Verification of Compton Effect: A beam of monochromatic X-ray of known wavelength λ is made incident on a graphite scatterer. These were observed at different angles with a Bragg spectrometer. Collimating Slits Scattered X ray Monochromatic X ray X ray Spectrometer Scatterer Figure 1.7 : Experimental verification of Compton Effect The intensity distribution verses wavelength for various angles is plotted as shown in figure. It is obvious that the curves have two peaks, one corresponding to modified radiations and the other to unmodified radiations. The difference between the two peaks on the wavelength axis gives the Compton shift ( λ). The curve shows the greater is the scattering angle; greater is the Compton shift in accordance with h 1 cos mo c Dr. Amita Maurya,, Asstt. Prof., Physics, People’s College of Research & Technology, Bhopal For more lecture notes visit http://sites.google.com/site/dramitamaurya 5 Unit I: Quantum Physics Lectures on Engineering Physics BE - 201 At θ = 90o, λ = 0.02426Å Unmodified radiation = 0o = 45o ∆λ Modified radiation λ’ = 90o ∆λ λ' = 135o ∆λ = 180o ∆λmax λ λ' λ Thus Compton Effect is experimentally verified. Dr. Amita Maurya,, Asstt. Prof., Physics, People’s College of Research & Technology, Bhopal For more lecture notes visit http://sites.google.com/site/dramitamaurya 6