Compton Shift Calculator

Choose incident input

Incident wavelength

Incident energy

Enter one type of incident value. The other is computed.

Incident wavelength

Common X‑ray: 0.071 nm for Mo Kα.

Unit mcmmmµmnmpmÅ

Incident energy

If wavelength mode is chosen, this is ignored.

Unit eVkeVMeVGeVJ

Scattering angle (θ)

Use 0 to 180 degrees, or 0 to π radians.

Unit deg rad

Output units

Δλ unit

mnmpmÅ

λ and λ′ unit

mnmpmÅ

Energy unit

eVkeVMeVGeVJ

Calculate Reset

Compton scattering relates the photon wavelength change to the scattering angle:

Once Δλ is known, the scattered wavelength and energy follow:

A direct energy form is also common: E′ = E / (1 + (E/(m_ec²))(1 − cos θ)).

1. Select whether you will enter incident wavelength or incident energy.
2. Enter the incident value and choose its unit.
3. Enter the scattering angle θ, then choose degrees or radians.
4. Pick output units for Δλ, wavelengths, and energies.
5. Click Calculate to view results above the form.

1) What the Compton Shift Represents

The Compton shift is the increase in photon wavelength after scattering from a free or weakly bound electron. It is direct evidence of momentum exchange between radiation and matter, and it is widely used in X‑ray and gamma‑ray physics. In practice, it links geometry (the scattering angle) to measurable spectral changes, which is why it appears in detector calibration, shielding studies, and backscatter analysis.

2) Key Constant: Electron Compton Wavelength

The scale of the effect is set by the electron Compton wavelength, λC = h/(mec). Numerically, λC ≈ 2.4263 pm. Your results display λC in meters to keep calculations traceable.

3) Angle Dependence and Physical Limits

The calculator applies Δλ = λC(1 − cos θ). At θ = 0°, the shift is zero. At θ = 180°, the maximum shift is Δλmax = 2λC ≈ 4.8526 pm. This bound is useful for quick sanity checks. A convenient reference point is θ = 90°, where cos θ = 0 and Δλ = λC ≈ 2.4263 pm.

4) Wavelength and Energy Inputs

You may enter either the incident wavelength or the incident photon energy. Internally, the tool converts values to SI units and uses E = hc/λ. For fast hand estimates, hc ≈ 1.239841984 keV·nm, so 12.4 keV corresponds to about 0.1 nm. The scattered energy can also be computed directly using E′ = E / (1 + (E/(mec²))(1 − cos θ)), where mec² ≈ 511 keV.

5) Typical Magnitudes in X‑ray Work

In the 0.05–0.2 nm range, picometer shifts are measurable. For example, at 0.071 nm and 90°, Δλ is about 2.426 pm, giving λ′ ≈ 0.0734 nm. That change reduces photon energy by a few percent, depending on geometry. At much higher energies (hundreds of keV), wavelengths are far smaller, but the shift limit stays the same; energy loss becomes more apparent in keV even though the wavelength change remains just a few picometers.

6) Using the Outputs in Experiments

If you measure a scattered wavelength λ′, compare it with the predicted λ + Δλ for your scattering angle. The scattered energy E′ helps set detector thresholds and estimate background from Compton continua. You can also estimate electron recoil energy as Ee ≈ E − E′, which is useful when interpreting Compton edges in scintillators and semiconductor detectors.

7) Unit Selection and Reporting

Use pm for Δλ to keep numbers readable. For wavelengths, nm or Å are common in lab reports. The result table supports clean copy‑paste, while CSV export supports structured logging.

8) Common Pitfalls and Checks

Keep θ within 0–180° (or 0–π radians) and ensure inputs are positive. Δλ should never exceed about 4.8526 pm. The consistency check compares two equivalent energy forms and should remain near 0% aside from rounding.

1) Does the Compton shift depend on the incident wavelength?

No. Δλ depends only on the scattering angle and λC. The incident wavelength changes the final wavelength and energy, but it does not change the shift itself.

2) Why is the maximum shift about 4.85 pm?

At 180° backscatter, the formula gives Δλ = 2λC. With λC ≈ 2.4263 pm, the maximum shift is approximately 4.8526 pm.

3) When should I enter energy instead of wavelength?

Use energy when your source, spectrum, or detector settings are specified in keV or MeV. Use wavelength when your work is reported in nm or Å, such as diffraction setups.

4) What does the energy consistency check mean?

The calculator computes E′ from λ′ and from the direct energy equation. The percent value shows their difference due to rounding. A near‑zero value indicates consistent results.

5) Can I apply this to bound electrons?

The standard equation assumes free electrons. Bound electrons can introduce small shifts and broadening. For many X‑ray energies, the free‑electron approximation remains a useful baseline.

6) Why does scattered energy decrease?

The photon transfers some energy and momentum to the recoiling electron. Because E = hc/λ, an increased wavelength corresponds to a decreased photon energy after scattering.

7) Which units are most practical for output?

Use pm for Δλ, and nm or Å for wavelengths. For energies, keV is typical for X‑rays and MeV for gamma rays. Pick units that keep values readable and comparable.