The Spectrum of Scattered X-Rays

Brad Roth
3 min readFeb 5, 2021

Chapter 15 of Intermediate Physics for Medicine and Biology discusses Compton Scattering. In this process, an x-ray photon scatters off a free electron, creating a scattered photon and a recoiling electron. The wavelength shift between the incident and scattered photons, Δλ, is given by the Compton scattering formula (Eq. 15.11 in IPMB)

Δλ = h/mc (1 − cosθ) ,

where h is Planck’s constant, c is the speed of light, m is the mass of the electron, and θ is the scattering angle. The quantity h/mc is called the Compton wavelength of the electron.

I enjoy studying experiments that first measure fundamental quantities like the Compton wavelength. Such an experiment is described in Arthur Compton’s article

Compton, A. H. (1923) The Spectrum of Scattered X-Rays. Physical Review, Volume 22, Pages 409–413.

Compton’s x-ray source (emitting the Kα line from molybdenum) irradiated a scatterer (graphite). He performed his experiment for different scattering angles θ. For each angle, he first collimated the scattered beam (using small holes in lead sheets) and then reflected it from a crystal (calcite). The purpose of the crystal was to determine the wavelength of the scattered photon by x-ray diffraction. Russ Hobbie and I don’t analyze x-ray diffraction in IPMB. The wavelength λ of the diffracted photon is given by Bragg’s law

λ = 2 d sinϕ ,

where d is the spacing of atomic planes (for calcite, d = 3.036 Å) and ϕ is the angle between the x-ray beam and the crystal surface. For fixed θ, Compton would rotate the crystal, thereby scanning ϕ and analyzing the beam as a function of wavelength. The intensity of the beam would be recorded by a detector (an ionization chamber).

Let’s analyze Compton’s experiment in a new homework problem.

Section 15.4

Problem 7½. Use the data below to calculate the Compton wavelength of the electron (in nm). Estimate the uncertainty in your value. Compton’s experiment detected x-rays at both the incident wavelength (coherent scattering) and at a modified or shifted wavelength (Compton scattering).

I like this exercise because it requires the reader to do many things:

  • Decide which spectral line is coherent scattering and which is Compton scattering.
  • Choose which angle θ to analyze.
  • Estimate the angle ϕ of each spectral peak from the data.
  • Approximate the uncertainty in the estimation of ϕ.
  • Convert the values of ϕ from degrees/minutes to decimal degrees.
  • Determine the wavelength for each angle using Bragg’s law.
  • Calculate the wavelength shift.
  • Relate the wavelength shift to the Compton wavelength.
  • Compute the Compton wavelength.
  • Propagate the uncertainty.
  • Convert from Ångstroms to nanometers.

If you can do all that, you know what you’re doing.

Compton’s experiment played a key role in establishing the quantum theory of light and wave-particle duality. He was awarded the 1927 Nobel Prize in Physics for this research. Let’s give him the last word. Here are the final two sentences of his paper.

This satisfactory agreement between the experiments and the theory gives confidence in the quantum formula for the change in wave-length due to scattering. There is, indeed, no indication of any discrepancy whatever, for the range of wave-length investigated, when this formula is applied to the wave-length of the modified ray.

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Brad Roth

Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.