Electrons with energy 80 keV are incident on the tungsten target of an X-ray tube. K-shell electrons

Q: Why do we use $12400$ or $12375$ in the wavelength formula?

The value $12400 \text{ eV \AA}$ (or $12375$ for higher precision) is the product of Planck's constant $h$ and the speed of light $c$. It is a handy shortcut to quickly convert between energy in electron-volts (eV) and wavelength in Angstroms (\AA).

Q: What happens if the incident energy is less than $72.5 \text{ keV}$?

If the incident energy is less than the K-shell binding energy, the incident electrons cannot knock out K-shell electrons. In that case, the characteristic K-series X-rays will not be emitted, though L-series or continuous X-rays might still be present.

Q: Why is there a minimum wavelength for the continuous spectrum?

The minimum wavelength corresponds to the maximum possible energy of an emitted X-ray photon. This happens when an incident electron loses all its kinetic energy in a single collision, converting it entirely into one photon.

Q: Does the continuous spectrum contain all wavelengths?

No, it contains all wavelengths from $\lambda_{\text{min}}$ up to infinity. Wavelengths shorter than $\lambda_{\text{min}}$ are physically impossible because they would require more energy than the incident electron possesses.

Q: Are characteristic X-rays always produced?

No, they are only produced if the accelerating voltage (incident electron energy) is sufficiently high to overcome the binding energy of the target's inner shell electrons.

Question

Electrons with energy $80 	ext{ keV}$ are incident on the tungsten target of an X-ray tube. K-shell electrons of tungsten have $72.5 	ext{ keV}$ energy. X-rays emitted by the tube contain only

Options:

(A) a continuous X-ray spectrum (Bremsstrahlung) with a minimum wavelength of $\approx 0.155 	ext{ \AA}$

(B) a continuous X-ray spectrum (Bremsstrahlung) with all wavelengths

(C) the characteristic X-ray spectrum of tungsten

(D) a continuous X-ray spectrum (Bremsstrahlung) with a minimum wavelength of $\approx 0.155 	ext{ \AA}$ and the characteristic X-ray spectrum of tungsten

Sigma Percentile · Accepted Answer

### Analyzing the Setup
When high-energy electrons strike a heavy target like tungsten, two distinct types of X-rays are produced. First, we have the **continuous X-rays**, also known as Bremsstrahlung, which are caused by the rapid deceleration of the incident electrons as they interact with the strong electric fields of the target nuclei. Second, we have the **characteristic X-rays**, which are emitted when an incident electron successfully knocks out an inner-shell electron (like one from the K-shell), and an outer electron falls in to fill the resulting vacancy.

### The Continuous Spectrum
Let's first look at the continuous spectrum. The maximum energy an X-ray photon can possess is exactly equal to the total kinetic energy of the incident electron. This maximum energy corresponds to the minimum wavelength, $\lambda_{	ext{min}}$, of the emitted X-rays. Using the standard energy-wavelength relation:

$$\lambda_{	ext{min}} = \frac{hc}{E_{	ext{incident}}}$$

Substituting the given incident energy of $80 	ext{ keV}$ (or $80 	imes 10^3 	ext{ eV}$) and using $hc \approx 12375 	ext{ eV \AA}$ for precision, we get:

$$\lambda_{	ext{min}} = \frac{12375}{80 	imes 10^3} \approx 0.155 	ext{ \AA}$$

This confirms that a continuous spectrum with a minimum wavelength of $\approx 0.155 	ext{ \AA}$ is indeed produced.

### The Characteristic Spectrum
Now, what about the characteristic X-rays? For these to be emitted, the incident electron must have enough energy to overcome the binding energy of an inner shell and eject an electron. The condition is strictly:

$$E_{	ext{incident}} > E_{	ext{binding}}$$

In our problem, the incident electrons have an energy of $80 	ext{ keV}$. The K-shell electrons of tungsten are bound with an energy of $72.5 	ext{ keV}$. Since $80 	ext{ keV} > 72.5 	ext{ keV}$, the incident electrons pack enough punch to eject the K-shell electrons!

Because the K-shell vacancy is created, outer shell electrons will drop down to fill it, emitting characteristic X-rays in the process. Therefore, the X-rays emitted by the tube will contain **both** a continuous spectrum with a minimum wavelength of $\approx 0.155 	ext{ \AA}$ and the characteristic X-ray spectrum of tungsten.

Animated Solution for Physics - Atoms and Nuclei: Electrons with energy $80 keV$ are incident on the tungsten target of an X-ray tube. K-shell electrons of tungsten have $72.5 keV$ energy. X-rays emitted by the tube contain only

Select Answer:

Visualized Solution

\text{X-ray Production Mechanisms}

\text{Minimum Wavelength of Continuous Spectrum}

\text{Condition for Characteristic X-rays}

\text{Comparing Energies}

\text{Conclusion}

The Sigma Insight: Bohr's Atomic Model and Energy Levels

Analyzing the Setup When high-energy electrons strike a heavy target like tungsten, two distinct types of X-rays are produced

The Continuous Spectrum Let's first look at the continuous spectrum

The Characteristic Spectrum Now, what about the characteristic X-rays? For these to be emitted, the incident electron must have enough energy to overcome the binding energy of an inner shell and eject an electron

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