# Total or kinetic energy in Bethe Bloch stopping power?

• I
• crick
In summary: They only differ by a constant value of ##mc^2##, which does not affect the derivative. In summary, the average loss of energy in a material per unit length of a particle can be described by the quantity ##dE/dx##. For ionization, the Bethe-Bloch formula is used, while for Bremmstralungh, the Bethe-Heilter formula is used. It is unclear if the energy in these formulas is the total relativistic energy or just the kinetic energy. However, it is generally accepted that the first case is more accurate. It is also unclear if a particle can change its mass while stopping in a material, as electrons are stable and do not decay. In calorimeters, the

#### crick

The average loss of energy in a material per unit length of a particle (in particular an electron, which is stable) is described by the quantity ##dE/dx##.

- for ionization it is given by the Bethe-Bloch formula $$-\left\langle {\frac {dE}{dx}}\right\rangle ={\frac {4\pi }{m_{e}c^{2}}}\cdot {\frac {nz^{2}}{\beta ^{2}}}\cdot \left({\frac {e^{2}}{4\pi \varepsilon _{0}}}\right)^{2}\cdot \left[\ln \left({\frac {2m_{e}c^{2}\beta ^{2}}{I\cdot (1-\beta ^{2})}}\right)-\beta ^{2}\right]$$
- for Bremmstralungh it is given by the Bethe-Heilter formula $${\displaystyle -\left\langle {\frac {dE}{dx}}\right\rangle \approx {\frac {4N_{a}Z^{2}\alpha ^{3}(\hbar c)^{2}}{m_{e}^{2}c^{4}}}E\ln {\frac {183}{Z^{1/3}}}}$$

I can't understand if the "energy ##E##" in the formulas is the total relativistic energy or the kinetic energy ##K## only?

In the first case ##E=K +mc^2##, while in the second case ##E=K##.

It looks like that the first case is the right one, since it's more general, but in that case I cannot understand how the particle, stopping in the material for various processes can change its mass (the electrons are stable so they do not decay after they have stopped). Does this really happen or does it loose before all its kinetic energy and then its rest energy?

I'm confused also because I read that in calorimeters the range of the particle is used to measure its energy: does this energy include the rest energy?

For the left side it does not matter as both only differ by a constant which doesn't change the derivative.
The Bethe-Heitler formula is only a good approximation for ##E\gg mc^2## which means it doesn't matter either.

crick
mfb said:
For the left side it does not matter as both only differ by a constant which doesn't change the derivative.
The Bethe-Heitler formula is only a good approximation for ##E\gg mc^2## which means it doesn't matter either.

Thank you for the answer! By "differing by a constant" in the left side are you referring to the fact that the rest mass ##mc^2## is constant?

It's simpler than that. df/dx = dg/dx is f(x) = g(x) + a constant.

Last edited:
crick said:
Thank you for the answer! By "differing by a constant" in the left side are you referring to the fact that the rest mass ##mc^2## is constant?
Right. The derivative of the total energy and the derivative of the kinetic energy are the same.

## 1. What is Bethe Bloch stopping power?

Bethe Bloch stopping power is a concept in physics that describes the rate at which a charged particle loses energy as it passes through a material. It is named after the scientists who first developed the theory, Hans Bethe and Felix Bloch.

## 2. How is total energy related to Bethe Bloch stopping power?

Total energy is directly related to Bethe Bloch stopping power because the amount of energy a particle loses is dependent on its initial total energy. As the particle loses energy, its stopping power decreases.

## 3. What is kinetic energy in the context of Bethe Bloch stopping power?

Kinetic energy is the energy that a charged particle possesses due to its motion. In the context of Bethe Bloch stopping power, it is the energy that is lost as the particle passes through a material and interacts with its atoms.

## 4. How is Bethe Bloch stopping power calculated?

Bethe Bloch stopping power is calculated using the Bethe-Bloch equation, which takes into account the properties of the material, the charge and mass of the particle, and its initial velocity. The equation can be derived from the principles of classical electrodynamics.

## 5. What factors can affect Bethe Bloch stopping power?

Several factors can affect Bethe Bloch stopping power, including the density and composition of the material, the charge and mass of the particle, and its initial velocity. Additionally, the energy loss can be affected by the electronic structure and binding energies of the atoms in the material.