Animated Solution for Physics - Dual Nature of Matter and Radiation: The X-ray beam coming from an X-ray tube will be

To truly understand the nature of an X-ray beam, we must first dive into the microscopic chaos that occurs inside an X-ray tube. Imagine a vacuum tube where a filament is heated until it boils off electrons. These electrons are then accelerated across a massive potential difference, $V$, gaining a tremendous amount of kinetic energy, $K=eV$.

When this high-speed stream of electrons violently crashes into a heavy metal target (like Tungsten), they don't just stop peacefully. They are rapidly decelerated by the strong electric fields of the target nuclei. In physics, whenever a charged particle accelerates or decelerates, it emits electromagnetic radiation. This specific type of radiation, born from the sudden braking of electrons, is called Bremsstrahlung (a German word literally meaning "braking radiation").

Maximum Energy

Now, let's play a game of probability. What is the absolute maximum energy a single X-ray photon can possess in this setup?

Consider the "luckiest" electron in the beam. This electron manages to avoid all minor deflections and crashes head-on into a nucleus, losing its entire kinetic energy in a single, catastrophic collision. By the law of conservation of energy, all of that lost kinetic energy is converted into exactly one X-ray photon.

Therefore, the maximum energy of the emitted photon is:

Minimum Wavelength

We know from Max Planck and Albert Einstein that the energy of a photon is inversely proportional to its wavelength, given by the famous equation:

Because there is a strict upper limit on the energy ($E_{\text{max}}$), there must be a strict lower limit on the wavelength. This is known as the cut-off wavelength or the Duane-Hunt limit (${\lambda }_{\text{min}}$).

It is physically impossible for the X-ray tube to emit a photon with a wavelength shorter than ${\lambda }_{\text{min}}$, because doing so would require the photon to have more energy than the electron that created it—a blatant violation of the conservation of energy!

A Tale of Many Collisions

But what about the billions of other electrons? The head-on collision we just described is statistically very rare. The vast majority of electrons will undergo multiple, glancing collisions as they penetrate the target.

In each of these minor collisions, an electron loses only a fraction of its kinetic energy. Consequently, the photons emitted from these glancing blows will have energies strictly less than $E_{\text{max}}$.

Since $E<E_{\text{max}}$, it mathematically guarantees that:

Because these energy losses are completely random and can take on any arbitrary value below the maximum, the resulting photons will cover an entire continuous spectrum of wavelengths. There is no theoretical upper limit to the wavelength (as an electron could theoretically lose an infinitesimally small amount of energy), meaning the spectrum extends towards infinity.

When we analyze the raw X-ray beam emerging from the tube, we do not see a single color (it is not monochromatic). Instead, we see a rich, continuous spectrum of radiation. However, this spectrum has a hard, impenetrable wall on the short-wavelength side.

Thus, the X-ray beam consists of all wavelengths larger than a certain minimum wavelength (${\lambda }_{\text{min}}$), making option (c) the undeniably correct choice.