# Nucleus Ratio of nucleons between surface and core

1. May 21, 2010

### dirtyhippy

1. The problem statement, all variables and given/known data

We suppose the nucleon density of a spherical nucleus where r<R1 is constant, and where R1<r<R2 the density linearly decreases to 0 at R2.

We call the surface nucleons (Number Ns) the number of nucleons contained in the volume R1<r<R2 and core nucleons (Nc) the number of nucleons in the volume r<R1.

It is easy to represent graphically the density (ρ) from 0 to R2 however,

to calculate the proportion of surface nucleons as a function of R2/R1 for Al27 and Po219 is proving more difficult.

2. Relevant equations

R1 = 1.1 A^1/3 -1.5
R2 = 1.1 A^1/3 +1.5

(the +/- 1.5 is not in the power!)

also

a^3-b^3 = (a-b).(a^2-b^2+ab)

and

a^4-b^4 = (a-b).(a^3-b^3+ab.(a+b))

3. The attempt at a solution

Nc + Ns = A

(ρ)c = Nc / V = 3Nc / 4π R1^3

(ρ)s = Ns / V = 3Ns / 4π ∫1/R^3 dR where the integral is from R1 to R2

Ns/N is the fraction being looked for and that is where we are stuck.

Were having problems relating Ns to R

If anyone could guide us through this it would be greatly appreciated.

Thanks

Ash + Aaron