| In a head-on collision between an incident electron and a nucleus, several energy-loss mechanisms can occur, depending on the energy of the electron and the nature of the interaction:
- Elastic Scattering:
- Coulomb Interaction: In an elastic collision, the incident electron interacts with the nucleus through Coulomb forces. Since the nucleus is much more massive than the electron, the energy transfer to the nucleus is minimal, but the direction of the electron's momentum is significantly altered.
- Rutherford Scattering: This type of elastic scattering is often referred to as Rutherford scattering, where the electron is deflected at large angles due to the intense electric field near the nucleus. If the deflection angle is close to 180 degrees (head-on), the electron may lose a significant portion of its kinetic energy, though not all of it is transferred to the nucleus due to the mass difference.
- Inelastic Scattering:
- Nuclear Excitation: If the incident electron has enough energy, it can excite the nucleus, causing it to move to a higher energy state. This process involves the electron losing some of its energy, which is absorbed by the nucleus.
- Bremsstrahlung (Radiation Loss): When an electron is decelerated in the intense electric field of the nucleus, it can emit X-rays or gamma rays. This radiation loss mechanism, known as Bremsstrahlung, is significant at higher electron energies and results in a reduction of the electron's kinetic energy.
- Energy Transfer to the Nucleus:
- Nuclear Recoil: The electron can transfer part of its energy to the nucleus, causing the nucleus to recoil. The amount of energy transferred depends on the incident energy of the electron and the angle of scattering. In a head-on collision, the energy transfer is maximized but is still limited by the mass ratio between the electron and the nucleus.
- Displacement Damage:
- Atom Displacement: In high-energy collisions, the energy transferred to the nucleus can be sufficient to displace atoms from their lattice positions, leading to defects in crystalline structures. This is particularly relevant in materials exposed to high radiation environments, such as in nuclear reactors or space applications.
- Radiative Processes:
- X-ray Production: The interaction between the electron and the nucleus can also lead to the emission of characteristic X-rays, depending on the atomic number of the nucleus. This is part of the inelastic scattering process where energy is lost through radiation.
The energy loss distribution in a head-on collision between an incident electron and a nucleus can be described by considering the kinematics of the collision and the principles of energy and momentum conservation. The distribution depends on the scattering angle and the nature of the interaction. One common equation used to describe the energy loss in such collisions is derived from Rutherford scattering theory and the Bethe-Bloch formula for energy loss by ionization.
For a head-on collision, the energy transfer is maximized. The maximum energy transfer Emax in such a collision can be approximated using the equation:
 ------------------------------------------------------------------------ [4686a]
where,
me
is the mass of the electron.
M is the mass of the nucleus.
γ is the Lorentz factor, accounting for relativistic effects, which is given by,
 --------------------------------------------------------------- [4686b]
c is the speed of light.
β is the velocity of the electron relative to the speed of light, which is given by v/c.
High-Energy Approximation: For high energies (i.e., when E0
is large compared to the rest energy of the electron),
γ becomes significant. However, me/M
is very small because the mass of the nucleus is much larger than the mass of the electron, so we can approximate:
 ---------------------------------------------------- [4686c]
Since me/M is very small, the denominator in Equation 4686a then is slightly greater than 1, but for a good approximation, we can treat it as approximately 1:
 ---------------------------------------------------- [4686d]
Since β ≈ 1 at high energies, we then have:
 ----------------------------------------------- [4686e]
Substituting  into Equation 4686d, then we have:
 ----------------------------------------------- [4686f]
Since M is the mass of the nucleus, and A is the atomic weight (where M = A×atomic mass unit (u)), the energy transfer depends inversely on the mass of the nucleus:
 ----------------------------------------------- [4686g]
The fundamental equation in Equation 4686a is based on the relativistic kinematics of an electron colliding with a nucleus. While this equation is derived from first principles and theoretically provides an accurate description of the maximum energy transfer in such a collision, it does not work well in practice due to the reasons below:
- Approximation and Simplifications:
- Mass Ratio: The fundamental equation assumes the mass ratio
me/M
between the electron and the nucleus is extremely small. However, in some cases, this assumption might oversimplify the interaction, especially for light nuclei where the mass difference isn't as extreme as it is for heavier nuclei.
- Relativistic Effects: The equation is relativistic, but it may not account for all the nuanced interactions that occur at high energies, particularly when dealing with very high-energy electrons or extreme relativistic speeds.
- Complexity and Real-World Factors:
- Experimental Conditions: In real experiments, various factors such as scattering angles, multiple collisions, inelastic interactions, and energy loss mechanisms (like Bremsstrahlung radiation) can influence the actual energy transfer, leading to deviations from the theoretical predictions.
- Material Properties: The fundamental equation doesn't consider specific material properties (e.g., electron density, binding energies, lattice effects) that could significantly impact energy transfer.
- Sensitivity to Parameters:
- The fundamental equation is sensitive to parameters like γ and β, which can vary significantly with changes in the incident electron's energy. Small errors in these parameters can lead to large discrepancies in the calculated energy loss, especially at relativistic speeds.
- The small mass of the electron relative to the nucleus means that even small changes in the relativistic factors can have a significant impact on the result.
- Assumption of Single Collision:
- The fundamental equation assumes a single, clean collision. In practice, electrons can undergo multiple interactions as they pass through a material, leading to energy loss distributions that differ from the idealized case.
Due to the reasons mentioned above,
empirical equations are often preferred over fundamental equations in real-world practice. An empirical equation is given by,
 ------------------------------------------- [4686h]
where,
- C1, C2, and
C3
are empirically determined constants that can be fitted to experimental data. Typical C1, C2, and
C3 are 2000, 1.01 and 0.15, respectively.
- C1: This constant scales the overall energy loss.
- C2 is a term which represents a small correction to account for relativistic effects.
- C3 is a correction term which reduces the impact at higher energies to minimize the deviation at different energy levels.
- The term
 adds a correction factor that accounts for energy-dependent behavior that is not well-captured by simpler linear terms.
This empirical equation in Equation 4686h is derived from experimental data and are designed to provide a close approximation of the desired outcome with less computational effort. They incorporate key factors and corrections into a simplified form that is more manageable in routine calculations. This makes them particularly useful in fields like electron microscopy, radiation physics, and materials science, where quick estimations are necessary for decision-making, design, and analysis.
The empirical equation predicts energy loss across a wide range of electron energies, from non-relativistic to relativistic regimes.
By applying Equations 4686h and 4686a, which is an empirical formula designed to estimate the maximum energy loss during a head-on collision between an electron and a nucleus, we can effectively plot the relationship between the speed of the incident electron and the maximum energy loss for different elements, such as Hydrogen (H), Carbon (C), Nitrogen (N), Oxygen (O), and Gold (Au). Figure 4686 visualize this relationship, represented by the curves plotted from the empirical equation, highlighting how different atomic masses influence the energy loss as the electron speed increases. In a collision, the amount of energy transferred to the nucleus depends on the mass of the nucleus relative to the electron. For lighter elements like Hydrogen, the maximum energy loss increases gradually with speed. That is, in collisions between incident electrons and matter, the energy transfer tends to be higher for light element matrices compared to heavy element matrices. In contrast, for heavy elements like gold, the nucleus is much heavier, so it is less easily moved by the colliding electron. As a result, the electron retains more of its energy after the collision, and less energy is transferred to the nucleus. Note that the curves plotted from the fundamental equation, in Figure 4686, overlap each other. This overlap occurs because the theoretical differences in energy loss between these elements, as predicted by the fundamental equation, are minimal.
Figure 4686. Maximum energy loss vs speed of incident electron for different elements. Here, Equations 4686h (Empirical) and 4686a (Fundamental) are applied. The speed of an electron with an energy of 200 keV in common TEM systems is approximately 2.08×10 8 m/s, which is about 69.6% of the speed of light. |
Similarly, in Electron Energy-Loss Spectroscopy (EELS) measurements, assuming the sample thicknesses are the same, carbon typically causes a higher background in the spectrum compared to gold because of the reasons below:
- Inelastic Scattering: EELS measures the energy lost by electrons as they pass through a material and interact with it. The background in an EELS spectrum is primarily due to inelastic scattering, where electrons lose energy by interacting with the electrons in the material.
- Plasmon Peaks and Background: Light elements like carbon have a higher density of low-energy plasmon peaks, which contribute to a higher inelastic scattering cross-section. These plasmons generate a broader background signal because of the more extended range of energy losses, including multiple scattering events.
- Atomic Number Effect: Gold, being a heavy element, has fewer inelastic scattering events at low energy losses compared to carbon. The scattering is more localized, and the energy loss tends to be more concentrated in specific, sharp features, such as inner-shell ionization edges, rather than contributing to a broad background.
At large incident electron energies (E 0), e.g. > 100 keV, the maximum energy transfer (E max) can exceed 1 eV, which is particularly relevant for light elements because this energy may be sufficient to displace atoms from their lattice sites.
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