Consider an alpha particle with kinetic energy E_ (alpha) which is incident on a material. Show that the distance traveled, ∆x, by the alpha particle after it has lost an amount of energy ∆E, in the material, can be found by the following equation:
∆x=Range (E_ (alpha)) - Range (E_ (alpha) - ∆E), where Range (E) = �ç (dE/SP), from zero to E
Assume that the stopping power, SP = (dE/SP)
To show that the distance traveled by the alpha particle after it has lost an amount of energy in the material can be found using the equation ∆x = Range(E_alpha) - Range(E_alpha - ∆E), we first need to understand some concepts related to the stopping power and range.
Stopping power (SP) refers to the rate at which a charged particle loses its kinetic energy as it passes through a material. It is usually denoted as (dE/dx), where dx represents the distance traveled. The stopping power depends on the energy of the particle.
Range (R) refers to the distance traveled by a charged particle in a material before it comes to a complete stop. In other words, it is the total distance over which the particle loses all of its kinetic energy. This range depends on the energy of the particle.
Now, let's derive the equation ∆x = Range(E_alpha) - Range(E_alpha - ∆E) using the concept of stopping power and range.
1. Start with the definition of range:
Range(E) = ∫(dE/SP) from E_alpha to E
This equation states that the range of an alpha particle with energy E_alpha is given by the integral of the inverse stopping power (dE/SP) with respect to energy E, starting from E_alpha. This integral sum represents the distance traveled by the alpha particle from its initial energy E_alpha till it loses all its kinetic energy (reaches stop).
2. Now, consider the distance traveled by the alpha particle when it loses an amount of energy ∆E.
Let's call this energy after the loss ∆E_reduced = E_alpha - ∆E.
So, the distance traveled by the alpha particle until it reaches the energy E_alpha - ∆E is given by:
Range(E_alpha - ∆E) = ∫(dE/SP) from E_alpha to E_alpha - ∆E
This represents the range from the initial energy E_alpha to the reduced energy E_alpha - ∆E.
3. Finally, the distance traveled (∆x) by the alpha particle after losing an energy ∆E is the difference in range:
∆x = Range(E_alpha) - Range(E_alpha - ∆E)
Using the above definitions and substitution, we arrive at the desired equation.
Therefore, the distance traveled (∆x) by the alpha particle after it has lost an amount of energy ∆E in the material can be found using the equation ∆x = Range(E_alpha) - Range(E_alpha - ∆E), where the range is given by Range(E) = ∫(dE/SP) from zero to E.