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:

Question

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)

Answer 1

To derive the equation ∆x = Range (E_ (alpha)) - Range (E_ (alpha) - ∆E), let's start by understanding the concept of range and stopping power.

In the context of this question, range represents the distance traveled by an alpha particle with a certain initial energy (E_ (alpha)) before it loses all of its energy. It can be calculated by integrating the stopping power (SP = dE/dx) from zero to the initial energy:

Range(E) = ∫ (0 to E) (dE/SP)

The stopping power (SP) represents the amount of energy (dE) lost per unit distance (dx) by the alpha particle as it passes through the material.

Now, let's consider the distance traveled (∆x) by the alpha particle after it has lost an amount of energy (∆E). We want to find an expression for ∆x.

We know that the original range (from zero to E_ (alpha)) represents the distance traveled by the alpha particle when it has not lost any energy (∆E = 0). So, we can denote this distance as:

Range(E_ (alpha)) = ∫ (0 to E_ (alpha)) (dE/SP)

Now, let's consider the range when the particle has lost an amount of energy ∆E. We need to find the new range by integrating the stopping power with respect to energy over the interval from zero to (E_ (alpha) - ∆E):

Range(E_ (alpha) - ∆E) = ∫ (0 to E_ (alpha) - ∆E) (dE/SP)

The distance traveled by the alpha particle after losing ∆E can be calculated by subtracting the new range from the original range:

∆x = Range (E_ (alpha)) - Range (E_ (alpha) - ∆E)
= ∫ (0 to E_ (alpha)) (dE/SP) - ∫ (0 to E_ (alpha) - ∆E) (dE/SP)
= ∫ (E_ (alpha) - ∆E to E_ (alpha)) (dE/SP)

So, based on the derivation, the equation to find the distance traveled (∆x) by the alpha particle after losing an amount of energy (∆E) in the material can be represented as:

∆x = Range (E_ (alpha)) - Range (E_ (alpha) - ∆E)

where Range (E) = ∫ (0 to E) (dE/SP).

Please note that the stopping power (SP) still needs to be defined or provided in order to compute the distance using this equation.