Question

A typical ionization energy - the energy needed to remove an electron- - for the elements is \(10 \mathrm{eV}\). Explain why the energy binding the electron to its atom can be ignored in Compton scattering involving an X-ray photon with wavelength about one-tenth of a nanometer.

Short answer

The energy of an X-ray photon, calculated with the given wavelength, significantly exceeds the ionization energy of the atoms (10 eV). Therefore, the energy binding the electron to its atom (ionization energy) can be ignored in Compton scattering scenarios, as the photon carries enough energy to ionize the electrons and still have plenty left for the scattering process.

Step-by-step solution

Calculate the energy of the X-ray photon

First, let's determine the energy of the incoming X-ray photon. One can use the Planck-Einstein relation \(E = h\nu\), where \(E\) is the energy, \(h\) is the Planck's constant (given in eV*s), and \(\nu\) is the frequency of the photon. However, the exercise provides the wavelength, \(\lambda\), of the X-ray photon, not its frequency. But, the relation between frequency and wavelength is \(\nu = \frac{c}{\lambda}\), where \(c\) is the speed of light in vacuum. Substituting this in the first equation gives \(E = \frac{hc}{\lambda}\). Substituting the given values for \(h = 4.1357 × 10^{-15} eV*s\), \(c = 3 × 10^8 m/s\), and \(\lambda = 0.1 × 10^{-9} m\) (converted from nanometer to meter) we find the energy of the X-ray photon.

Compare ionization energy with X-ray photon energy

Now, compare the calculated X-ray photon's energy with the given ionization energy of the elements (10 eV). If the X-ray photon's energy significantly exceeds the ionization energy, it means the binding energy can be ignored during the Compton scattering process, because the photon carries enough energy to ionize the electrons and still have plenty left for the scattering process.

Conclusion

Based on the comparison, one can conclude why the energy binding the electron to its atom can be ignored during the Compton scattering involving an X-ray photon with the given wavelength. Note that in general terms, the energy of an X-ray photon typically orders of magnitude higher than an atom's ionization energy, hence justifying the assumption in the context of Compton scattering.

Key concepts

Ionization Energy

Ionization energy is a key concept when discussing how electrons are bound to atoms. It represents the minimal energy necessary to remove an electron from the outer shell of an atom. Expressed in electron volts (eV), it varies depending on the specific element and electronic environment.
For many elements, typical ionization energies are around 10 eV. This value shows how tightly an electron is bound to an atom.
Despite this, in processes involving very high-energy particles, such as X-ray photons in Compton scattering, the ionization energy becomes less significant. This is because such interactions involve energies far superior to the energies binding electrons to their atoms, allowing high-energy photons to overcome these bindings with ease.
In summary, the ionization energy can often be ignored when dealing with high-energy photons due to the vast energy difference.

X-ray Photon Energy

X-ray photon energy is a crucial element of Compton scattering calculations. It is directly tied to the photon's wavelength, offering insight into how much energy the photon carries.
The energy of an X-ray photon can be determined using the relation \[ E = \frac{hc}{\lambda} \], where \(h\) is the Planck constant, \(c\) is the speed of light, and \(\lambda\) corresponds to the wavelength of the photon.
X-rays generally have very short wavelengths, around one-tenth of a nanometer in this scenario.
This short wavelength means they carry very high energy.
Consequently, the energy of an X-ray photon often vastly exceeds typical ionization energies, making them significant players in interactions that involve considerable energy transfers, such as those in Compton scattering.

Planck-Einstein Relation

The Planck-Einstein relation acts as a bridge between the particle and wave characteristics of photons. It provides a method to calculate a photon's energy based on its frequency. The relation is defined as \[ E = hu \] where \(E\) is the energy, \(u\) is the frequency, and \(h\) is the Planck constant.
This relation is pivotal in quantum mechanics, emphasizing the dual nature of light as both wave and particle. It illustrates how electromagnetic waves like light, radio waves, and X-rays carry energy in quantized amounts, called photons.
When wavelength, \(\lambda\), is provided instead of frequency, we can also use \[ u = \frac{c}{\lambda} \] substituting it into the original relation to find the energy of the photon in terms of wavelength, as shown in the step-by-step solution.
Understanding the Planck-Einstein relation is necessary for analyzing phenomena like Compton scattering, as it allows us to explore how X-ray photons interact with matter.