Estimating the energy of an alpha particle using Bethe's formula

[PandaKitten]

Homework Statement:

Full question is below

Relevant Equations:

Bethe's formula

The question is below. I tried reasoning that because x is constant, E is also constant however that gives me values in the range of 10^51. Then I tried to use numpy's ivp_solve function to solve the differential equation however I wasn't able to get that working either. Apparently I'm meant to use Taylor's expansion to estimate E however I'm not sure how I would do that.

 

[Charles Link]

See https://en.wikipedia.org/wiki/Bethe...rmula or Bethe,stopping power of the material).
You have a couple of terms in your formula that seem to be inconsistent with the formula of the "link".

 

[PandaKitten]

See https://en.wikipedia.org/wiki/Bethe_formula#:~:text=The Bethe formula or Bethe,stopping power of the material).
You have a couple of terms in your formula that seem to be inconsistent with the formula of the "link".

This is a simplified version which we were given because the alpha particle is non relativistic so beta << 1 and it also uses reduced mass mu and also E = 1/2 m_e/m_He.

 

[Charles Link]

This is a simplified version which we were given because the alpha particle is non relativistic so beta << 1 and it also uses reduced mass mu and also E = 1/2 m_e/m_He.

This last statement is unclear. Please show what $\mu$ is algebraically. Supposedly $E$ is the energy of the alpha particle. This looks like a case where your notes that you might have taken in lecture contain errors.

 

[PandaKitten]

This is what the question states

 

[Charles Link]

I have to admit, I can't make sense out of their asking for a graph of $-\frac{dE}{dP}$ vs. $E$. Perhaps someone else can see what they are asking for here.

 

[kdv]

Homework Statement:: Full question is below
Relevant Equations:: Bethe's formula

The question is below. I tried reasoning that because x is constant, E is also constant however that gives me values in the range of 10^51. Then I tried to use numpy's ivp_solve function to solve the differential equation however I wasn't able to get that working either. Apparently I'm meant to use Taylor's expansion to estimate E however I'm not sure how I would do that.
View attachment 282789

Are you 100% sure they are asking a graph of -dE/dP vs E?? I would be willing to bet that this is a typo and they really are just asking a plot of -dE/dx vs E, which is then simple, they just want the graph of the expression they provided. (By the way, their definition of reduced mass is very unorthodox!)

 

[hutchphd]

I have to admit, I can't make sense out of their asking for a graph of −dEdP vs. E. Perhaps someone else can see what they are asking for he

For instance, one might wish to choose a pressure where the response of the detector has particular characteristics for a certain energy alpha particle. Maybe there is a nice flat spot or an edge.

 

[Vanadium 50]

dE/dP makes no sense on its own. d²E/dPdx means something.

 

[hutchphd]

They were asking in the context of a 10cm path length so I assume that is the detector path length and an average energy if I recall
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