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

In summary, the question asks about using Bethe's formula to estimate the energy of an alpha particle, but there are inconsistencies in the given formula. The student tried to reason using constant values, but the resulting values were in the range of 10^51. They also attempted to use numpy's ivp_solve function, but were unsuccessful. The solution may involve using Taylor's expansion, but the student is unsure how to proceed. The question also mentions a graph of -dE/dP vs E, which may be a typo and could potentially be -dE/dx vs E.
  • #1
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.
AlphaHelp.png
 
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  • #4
PandaKitten said:
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.
 
  • #5
This is what the question states
daww.png
 
  • #6
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.
 
  • #7
PandaKitten said:
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!)
 
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  • #8
Charles Link said:
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.
 
  • #9
dE/dP makes no sense on its own. d2E/dPdx means something.
 
  • #10
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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