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

[gennarakis]

Homework Statement

Knowing the Z (number of protons) of an element, how can I find N (number of neutrons) or A (mass number) of the element?

Homework Equations

I tried to use the semi-empirical mass formula E_B = α_νΑ - α_sA^2/3 - α_cZ²/A^1/3 - α_Α(Α-2Ζ)²/Α - δ(Α,Ζ) for this.

The Attempt at a Solution

I[/B] tried to to find the maximum binding energy by dE_b/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 A^4/3 + 352.16 A^2/3 + 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.

 

[SteamKing]

Homework Statement

Knowing the Z (number of protons) of an element, how can I find N (number of neutrons) or A (mass number) of the element?

Homework Equations

I tried to use the semi-empirical mass formula E_B = α_νΑ - α_sA^2/3 - α_cZ²/A^1/3 - α_Α(Α-2Ζ)²/Α - δ(Α,Ζ) for this.

The Attempt at a Solution

I[/B] tried to to find the maximum binding energy by dE_b/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 A^4/3 + 352.16 A^2/3 + 67665.92 = 0

Now I can't find a way to solving for A.

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

Only way to find A that I can see is by substituting numerically values for A and see when the equation is valid.

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.

 

[gennarakis]

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..

 

[SteamKing]

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..

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.