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Find the mass number A of the most stable nuclei given Z

  1. Feb 7, 2015 #1
    1. The problem statement, all variables and given/known data
    Knowing the Z (number of protons) of an element, how can I find N (number of neutrons) or A (mass number) of the element?

    2. Relevant equations
    I tried to use the semi-empirical mass formula EB = ανΑ - αsA2/3 - αcZ2/A1/3 - αΑ(Α-2Ζ)2/Α - δ(Α,Ζ) for this.

    3. The attempt at a solution
    I
    tried to to find the maximum binding energy by dEb/dA=0 for a given Z number. I came to the following equation where I substituted Z=32 as an example for Germanium:

    2113.9 A2 - 11.4 A4/3 + 352.16 A2/3 + 67665.92 = 0

    Now I can't find a way to solving for A. Only way to find A that I can see is by substituting numerically values for A and see when the equation is valid.
     
  2. jcsd
  3. Feb 7, 2015 #2

    SteamKing

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    This is not surprising, since the equation contains fractional exponents on the unknown variable A.

    And your problem with this is what exactly?

    You can also graph the equation, or use a numerical technique like Newton's method to find solutions.
     
  4. Feb 7, 2015 #3
    This is an exams problem and you have a limited amount of time to solve it.

    I suppose there isn't an exact method to solve for these fractional exponents..

    Can you give me a hint for Newton's method..
     
  5. Feb 7, 2015 #4

    SteamKing

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    Newton's method has many different references:

    http://en.wikipedia.org/wiki/Newton's_method

    Since you are concerned about solving this equation during an exam, you can always use an iterative approach. After all, you are starting with a given Z, from which you can estimate a reasonable trial value of A for that particular element. The trial value would also work as an initial guess for Newton's method.
     
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