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We start with rearranging the terms in [link] and squaring it:

[ ( E f E ˜ f ) + m 0 c 2 ] 2 = E 2 .

In the next step, we substitute [link] for E 2 , simplify, and divide both sides by c 2 to obtain

( E f / c E ˜ f / c ) 2 + 2 m 0 c ( E f / c E ˜ f / c ) = p 2 .

Now we can use [link] to express this form of the energy equation in terms of momenta. The result is

( p f p ˜ f ) 2 + 2 m 0 c ( p f p ˜ f ) = p 2 .

To eliminate p 2 , we turn to the momentum equation [link] , rearrange its terms, and square it to obtain

( p f p ˜ f ) 2 = p 2 and ( p f p ˜ f ) 2 = p f 2 + p ˜ f 2 2 p f · p ˜ f .

The product of the momentum vectors is given by [link] . When we substitute this result for p 2 in [link] , we obtain the energy equation that contains the scattering angle θ :

( p f p ˜ f ) 2 + 2 m 0 c ( p f p ˜ f ) = p f 2 + p ˜ f 2 2 p f p ˜ f cos θ .

With further algebra, this result can be simplified to

1 p ˜ f 1 p f = 1 m 0 c ( 1 cos θ ) .

Now recall [link] and write: 1 / p ˜ f = λ / h and 1 / p f = λ / h . When these relations are substituted into [link] , we obtain the relation for the Compton shift:

λ λ = h m 0 c ( 1 cos θ ) .

The factor h / m 0 c is called the Compton wavelength    of the electron:

λ c = h m 0 c = 0.00243 nm = 2.43 pm .

Denoting the shift as Δ λ = λ λ , the concluding result can be rewritten as

Δ λ = λ c ( 1 cos θ ) .

This formula for the Compton shift describes outstandingly well the experimental results shown in [link] . Scattering data measured for molybdenum, graphite, calcite, and many other target materials are in accord with this theoretical result. The nonshifted peak shown in [link] is due to photon collisions with tightly bound inner electrons in the target material. Photons that collide with the inner electrons of the target atoms in fact collide with the entire atom. In this extreme case, the rest mass in [link] must be changed to the rest mass of the atom. This type of shift is four orders of magnitude smaller than the shift caused by collisions with electrons and is so small that it can be neglected.

Compton scattering is an example of inelastic scattering    , in which the scattered radiation has a longer wavelength than the wavelength of the incident radiation. In today’s usage, the term “Compton scattering” is used for the inelastic scattering of photons by free, charged particles. In Compton scattering, treating photons as particles with momenta that can be transferred to charged particles provides the theoretical background to explain the wavelength shifts measured in experiments; this is the evidence that radiation consists of photons.

Compton scattering

An incident 71-pm X-ray is incident on a calcite target. Find the wavelength of the X-ray scattered at a 30 ° angle. What is the largest shift that can be expected in this experiment?

Strategy

To find the wavelength of the scattered X-ray, first we must find the Compton shift for the given scattering angle, θ = 30 ° . We use [link] . Then we add this shift to the incident wavelength to obtain the scattered wavelength. The largest Compton shift occurs at the angle θ when 1 cos θ has the largest value, which is for the angle θ = 180 ° .

Solution

The shift at θ = 30 ° is

Δ λ = λ c ( 1 cos 30 ° ) = 0.134 λ c = ( 0.134 ) ( 2.43 ) pm = 0.325 pm .

This gives the scattered wavelength:

λ = λ + Δ λ = ( 71 + 0.325 ) pm = 71.325 pm .

The largest shift is

( Δ λ ) max = λ c ( 1 cos 180 0 ) = 2 ( 2.43 pm ) = 4.86 pm .

Significance

The largest shift in wavelength is detected for the backscattered radiation; however, most of the photons from the incident beam pass through the target and only a small fraction of photons gets backscattered (typically, less than 5%). Therefore, these measurements require highly sensitive detectors.

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Check Your Understanding An incident 71-pm X-ray is incident on a calcite target. Find the wavelength of the X-ray scattered at a 60 ° angle. What is the smallest shift that can be expected in this experiment?

( Δ λ ) min = 0 m at a 0 ° angle; 71.0 pm + 0.5 λ c = 72. 215 pm

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Summary

  • In the Compton effect, X-rays scattered off some materials have different wavelengths than the wavelength of the incident X-rays. This phenomenon does not have a classical explanation.
  • The Compton effect is explained by assuming that radiation consists of photons that collide with weakly bound electrons in the target material. Both electron and photon are treated as relativistic particles. Conservation laws of the total energy and of momentum are obeyed in collisions.
  • Treating the photon as a particle with momentum that can be transferred to an electron leads to a theoretical Compton shift that agrees with the wavelength shift measured in the experiment. This provides evidence that radiation consists of photons.
  • Compton scattering is an inelastic scattering, in which scattered radiation has a longer wavelength than that of incident radiation.

Conceptual questions

Discuss any similarities and differences between the photoelectric and the Compton effects.

Answers may vary

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Which has a greater momentum: an UV photon or an IR photon?

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Does changing the intensity of a monochromatic light beam affect the momentum of the individual photons in the beam? Does such a change affect the net momentum of the beam?

no; yes

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Can the Compton effect occur with visible light? If so, will it be detectable?

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Is it possible in the Compton experiment to observe scattered X-rays that have a shorter wavelength than the incident X-ray radiation?

no

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Show that the Compton wavelength has the dimension of length.

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At what scattering angle is the wavelength shift in the Compton effect equal to the Compton wavelength?

right angle

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Problems

What is the momentum of a 589-nm yellow photon?

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What is the momentum of a 4-cm microwave photon?

1.66 × 10 32 kg · m/s

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In a beam of white light (wavelengths from 400 to 750 nm), what range of momentum can the photons have?

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What is the energy of a photon whose momentum is 3.0 × 10 −24 kg · m/s ?

56.21 eV

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What is the wavelength of (a) a 12-keV X-ray photon; (b) a 2.0-MeV γ -ray photon?

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Find the momentum and energy of a 1.0-Å photon.

6.63 × 10 23 kg · m/s ; 124 keV

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Find the wavelength and energy of a photon with momentum 5.00 × 10 −29 kg · m/s .

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A γ -ray photon has a momentum of 8.00 × 10 −21 kg · m/s . Find its wavelength and energy.

82.9 fm; 15 MeV

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(a) Calculate the momentum of a 2.5 - µ m photon. (b) Find the velocity of an electron with the same momentum. (c) What is the kinetic energy of the electron, and how does it compare to that of the photon?

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Show that p = h / λ and E f = h f are consistent with the relativistic formula E 2 = p 2 c 2 + m 0 2 c 2 .

(Proof)

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Show that the energy E in eV of a photon is given by E = 1.241 × 10 −6 eV · m / λ , where λ is its wavelength in meters.

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For collisions with free electrons, compare the Compton shift of a photon scattered as an angle of 30 ° to that of a photon scattered at 45 ° .

Δ λ 30 / Δ λ 45 = 45.74 %

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X-rays of wavelength 12.5 pm are scattered from a block of carbon. What are the wavelengths of photons scattered at (a) 30 ° ; (b) 90 ° ; and, (c) 180 ° ?

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Practice Key Terms 8

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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