Free Compton Scattering Calculator

What Is the Compton Effect?

The Compton effect is a fundamental light–matter interaction where a photon’s wavelength increases and its direction changes after colliding with a particle (typically a free or loosely bound electron). This phenomenon provides direct evidence for the particle nature of light, as it can be accurately modeled as an elastic collision that conserves both energy and momentum.

Using the Compton Scattering Calculator

The Compton Scattering Calculator—also referred to as a Compton Effect Calculator, Photon Scattering Calculator, Wavelength Extension Calculator, or Delta Lambda Calculator—enables you to quickly determine the wavelength shift ($\Delta\lambda$) that a photon experiences when it scatters through a given angle. By entering the scattering angle and the rest mass of the target particle (e.g., an electron), the tool computes the resulting wavelength extension using the standard Compton scattering formula.

The Scattering Process

Although a photon has zero rest mass, it carries energy proportional to its frequency. When this photon strikes a nearly free electron, the interaction resembles a classical elastic collision: the total energy and momentum of the system remain unchanged. Because the electron’s binding energy is negligible compared with typical X‑ray photon energies, the incident photon supplies virtually all the initial energy. This simplification is why the Compton effect is especially prominent in the X‑ray region of the electromagnetic spectrum.

The Compton Scattering Formula

The wavelength shift predicted by Compton’s theory is:

$$\Delta \lambda = \frac{h}{m c} (1 - \cos\theta)$$

where:

• $h = 6.62607 \times 10^{-34}\ \text{J·s}$ is Planck’s constant,
• $m$ is the rest mass of the scattering particle,
• $c = 299\,792\,458\ \text{m/s}$ is the speed of light,
• $\theta$ is the angle between the incident and scattered photon directions.

The factor $\frac{h}{m c}$ is known as the Compton wavelength of the particle. For an electron, the Compton wavelength equals $2.426\ \text{pm}$ (picometers). The same quantity appears in any Compton Wavelength Calculator.

How the Scattering Angle Affects the Shift

The value of $\Delta\lambda$ depends strongly on the scattering angle:

• At $\theta = 0^\circ$, $\cos\theta = 1$ and $\Delta\lambda = 0$ – no scattering occurs.
• At $\theta = 180^\circ$, $\cos\theta = -1$ and $\Delta\lambda = 2 \times \frac{h}{m c}$ – the maximum possible shift, equal to twice the Compton wavelength.

Thus, the largest energy loss (and greatest wavelength increase) happens when the photon is scattered directly backward.

Practical Example: Scattering off a Free Electron

Suppose a photon scatters from a free electron at an angle of $\theta = 80^\circ$. Taking the electron rest mass $m_e = 9.109 \times 10^{-31}\ \text{kg}$, the electron’s Compton wavelength is:

$$\frac{h}{m_e c} \approx 2.426\ \text{pm}$$

With $\cos 80^\circ \approx 0.1736$, the wavelength shift becomes:

$$\Delta \lambda = 2.426\ \text{pm} \times (1 - 0.1736) \approx 2.005\ \text{pm}$$

Now consider two photons with different initial energies:

• A 2 eV photon (red‑orange visible light) has a wavelength of about 620 nm.
• A 20 keV photon (hard X‑ray) has a wavelength of about 62 pm.

Applying the same shift of $2.005\ \text{pm}$:

• For the visible photon, the relative change is only $\frac{2.005\ \text{pm}}{620\ \text{nm}} \approx 0.000323\%$, which is negligible in practice.
• For the X‑ray photon, the relative change is $\frac{2.005\ \text{pm}}{62\ \text{pm}} \approx 3.23\%$, a substantial fraction that measurably alters the photon’s energy and direction.

When Does Compton Scattering Matter Most?

As the example shows, Compton scattering is practically irrelevant for visible and longer‑wavelength light because the shift is tiny compared with the original wavelength. It becomes highly significant for X‑rays and gamma rays, where the Compton shift represents a sizable percentage of the initial wavelength. This tool is therefore most useful in high‑energy physics, medical imaging (X‑ray and gamma‑ray diagnostics), and astrophysics.