Nucleus Ratio of nucleons between surface and core

[dirtyhippy]

Homework Statement

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.

Homework 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))

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

 

[mark726]

Hello Ash and Aaron,

Thank you for your post. It seems like you are trying to calculate the proportion of surface nucleons as a function of R2/R1 for two specific elements, Al27 and Po219. To do this, you will need to use the equations you have provided and some basic concepts from nuclear physics.

First, let's define some variables:
- A = atomic mass number (number of nucleons) of the nucleus
- R1 = radius of the core (where the density is constant)
- R2 = radius of the surface (where the density decreases to 0)
- Nc = number of core nucleons
- Ns = number of surface nucleons
- ρc = density of the core
- ρs = density of the surface

Now, let's use the equations you have provided to relate these variables:
- R1 = 1.1 A^(1/3) - 1.5
- R2 = 1.1 A^(1/3) + 1.5
- Nc + Ns = A
- ρc = 3Nc / (4π R1^3)
- ρs = 3Ns / (4π ∫1/R^3 dR) (Note: I believe there is a typo in this equation and it should be ∫1/R^2 dR instead of ∫1/R^3 dR)

Now, let's use these equations to calculate the proportion of surface nucleons (Ns/N) as a function of R2/R1.

For Al27:
- A = 27
- R1 = 1.1(27^(1/3)) - 1.5 = 3.6 fm
- R2 = 1.1(27^(1/3)) + 1.5 = 5.2 fm
- ρc = 3Nc / (4π (3.6)^3) = 0.044 Nc
- ρs = 3Ns / (4π (5.2)^2) = 0.028 Ns

To calculate Ns/N, we need to find Ns and N. We can do this by solving the equations Nc + Ns = A and ρc = Nc / V, where V is the volume of the nucleus (4/