Question

In a Compton scattering experiment, a photon with a wavelength ?=1.50x10^-3 nm collide with a stationary electron. After the collision, the electron recoils at 0.500c a) What is the energy and wavelength of the scattered photon? b) through what angle with respect to the incident direction was the photon scattered? [Hint: Me=0.511 MeV/c² or Me=9.11x10-31 kg]

Answer 1

In order to answer this question requires conservation of energy, the energy of a photon, the relativistic energy of a particle, and the equation for Compton scattering.

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Part (a)

Step 1) To find the energy and wavelength of the scattered photon, first write out the conservation of energy between the photon and electron:

That shows that the sum of the initial energies of the gamma ray and the electron has to equal the final energies of them.

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Step 2) Fill in expressions for the intial energy of the photon and electron (using the rest energy for the initial energy of the electron), as well as the final (relativistic) energy of the electron:

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Step 3) Rearrange that to solve for the final photon energy:

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Step 4) Fill in 1.24 keV*nm for the product of Planck's constant h and the speed of light c, 1.5*10^-3 nm for the initial photon wavelength , 511keV/c² (the 0.511 MeV/c²) for the mass of the electron m, and 0.500c for the velocity v of the electron after the collision:

So the final energy of the photon is about 748 keV.

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Step 5) To find the wavelength of the photon, start with the relationship between the energy E and wavelength of a photon:

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Step 6) Rearrange that to find the wavelength:

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Step 7) Fill in 1.24 keV*nm for the product hc, and 747.6 keV for the energy E of the photon that was just found:

So the wavelength of the photon after the collision is 0.00166 nm (or 1.66*10^-3 nm).

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Part (b)

Step 1) To find the angle through which the photon was scattered, start with the Compton scattering equation:

with being the wavelength of the reflected photon, the initial wavelength, h for Planck's constant, m for the mass of the electron, c for the speed of light, and for the angle that the photon scatters through.

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Step 2) To simplify calculating it later, multiply the right side by c/c:

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Step 3) Rearrange that to solve for the angle:

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Step 4) Fill in 1.24 keV*nm for the product hc, 511 keV/c² for the mass of the electron m, 1.66*10^-3 nm for the new wavelength , and 1.50*10^-3 nm for the original wavelength :

So the angle that the photon scatters through in the collision is 20.9^o.