The experimental data in this figure are plotted in arbitrary units so that the height of the profile reflects the intensity of the scattered beam above background noise. Compton Shift As given by Compton, the explanation of the Compton shift is that in the target material, graphite, valence electrons are loosely bound in the atoms and behave like free electrons. Compton assumed that the incident X-ray radiation is a stream of photons.
An incoming photon in this stream collides with a valence electron in the graphite target. In the course of this collision, the incoming photon transfers some part of its energy and momentum to the target electron and leaves the scene as a scattered photon.
This model explains in qualitative terms why the scattered radiation has a longer wavelength than the incident radiation. Put simply, a photon that has lost some of its energy emerges as a photon with a lower frequency, or equivalently, with a longer wavelength. To show that his model was correct, Compton used it to derive the expression for the Compton shift. In his derivation, he assumed that both photon and electron are relativistic particles and that the collision obeys two commonsense principles: 1 the conservation of linear momentum and 2 the conservation of total relativistic energy.
In the following derivation of the Compton shift, and denote the energy and momentum, respectively, of an incident photon with frequency f. Immediately after the collision, the outgoing photon has energy momentum and frequency The direction of the incident photon is horizontal from left to right, and the direction of the outgoing photon is at the angle as illustrated in Figure. The scattering angle is the angle between the momentum vectors and and we can write their scalar product:.
This assumption is valid for weakly bound electrons that, to a good approximation, can be treated as free particles. Our first equation is the conservation of energy for the photon-electron system:. The left side of this equation is the energy of the system at the instant immediately before the collision, and the right side of the equation is the energy of the system at the instant immediately after the collision.
Our second equation is the conservation of linear momentum for the photon—electron system where the electron is at rest at the instant immediately before the collision:. The left side of this equation is the momentum of the system right before the collision, and the right side of the equation is the momentum of the system right after collision. The entire physics of Compton scattering is contained in these three preceding equations——the remaining part is algebra.
We start with rearranging the terms in Figure and squaring it:. In the next step, we substitute Figure for simplify, and divide both sides by to obtain. Now we can use Figure to express this form of the energy equation in terms of momenta. The result is. To eliminate we turn to the momentum equation Figure , rearrange its terms, and square it to obtain.
The product of the momentum vectors is given by Figure. When we substitute this result for in Figure , we obtain the energy equation that contains the scattering angle. Now recall Figure and write: and When these relations are substituted into Figure , we obtain the relation for the Compton shift:.
The factor is called the Compton wavelength of the electron:. Denoting the shift as the concluding result can be rewritten as. This formula for the Compton shift describes outstandingly well the experimental results shown in Figure. Scattering data measured for molybdenum, graphite, calcite, and many other target materials are in accord with this theoretical result. The nonshifted peak shown in Figure 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 Figure 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 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 pm X-ray is incident on a calcite target. Find the wavelength of the X-ray scattered at a 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, We use Figure. Then we add this shift to the incident wavelength to obtain the scattered wavelength. The largest Compton shift occurs at the angle when has the largest value, which is for the angle.
Solution The shift at is. Therefore, these measurements require highly sensitive detectors. Check Your Understanding An incident pm X-ray is incident on a calcite target.
What is the smallest shift that can be expected in this experiment? Discuss any similarities and differences between the photoelectric and the Compton effects. 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? Can the Compton effect occur with visible light?
If so, will it be detectable? Is it possible in the Compton experiment to observe scattered X-rays that have a shorter wavelength than the incident X-ray radiation? At what scattering angle is the wavelength shift in the Compton effect equal to the Compton wavelength? In a beam of white light wavelengths from to nm , what range of momentum can the photons have?
What is the energy of a photon whose momentum is? What is the wavelength of a a keV X-ray photon; b a 2. Find the wavelength and energy of a photon with momentum. A -ray photon has a momentum of Find its wavelength and energy.
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Goel, A. Compton effect. Reference article, Radiopaedia. URL of Article. Probability of Compton effect directly proportional to number of outer shell electrons, i. Quiz questions.
Simpkin D. L'Annunziata MF. Elsevier Science. Read it at Google Books - Find it at Amazon 3. Walter Huda, Richard M. Review of Radiologic Physics.
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