Nuclear Models — Question 4

Question 4

Competition between Coulomb and asymmetry energies favors neutron excess

At fixed A, ignore neutron-proton rest-mass difference and pairing. Approximate the Z-dependent mass energy by $F(Z)=a_cZ^2A^{-1/3}+a_a(A-2Z)^2/A$. Treat Z as continuous, with $a_c=.700\,\mathrm{MeV}$ and $a_a=23.0\,\mathrm{MeV}$.

1. Minimize F to derive the preferred Z.

2. Evaluate it for A=100 and infer the approximate neutron-to-proton ratio. Check that the stationary point is a minimum.

Question 4 - Solution

Competition between Coulomb and asymmetry energies favors neutron excess

(a) Method and calculation

$$F\prime(Z)=2a_cZA^{-1/3}-4a_a(A-2Z)/A=0.$$$$\boxed{Z_* =\frac{A}{2+[a_c/(2a_a)]A^{2/3}}}.$$

(b) Interpretation and further calculation

$$\boxed{Z_*(100)=42.96,\quad N_*=57.04,\quad N_*/Z_*=1.328}.$$$$\boxed{F\prime\prime(Z)=2a_cA^{-1/3}+8a_a/A>0}.$$

Physical check

Coulomb repulsion shifts the minimum below Z=A/2. Integer Z, pairing, shell structure and neutron-proton mass difference refine the actual stability pattern.