In the Fermi-gas model

PHYSICS · Multi-part · Fermi University Physics · Nuclear Physics · Nuclear constituents, size and binding · The nuclear force and nuclear models

The Nucleus Resolved, Ch The shell model: a mean field, a spin-orbit term and the magic numbers

Problem

In the Fermi-gas model, protons and neutrons form two independent, symmetric Fermi gases confined to the nuclear volume. For

q

nucleons of one kind in that volume, the Fermi energy scales as

\epsilon_F(q)\propto q^{2/3}

, and the mean kinetic energy per nucleon is

\frac{3}{5}\epsilon_F(q)

. Write

\epsilon_F(q) = \epsilon_F\left(\frac{2q}{A}\right)^{2/3},

where

\epsilon_F

is the Fermi energy of symmetric matter (

Z=N=A/2

) at the same density. ((a)) Write the total kinetic energy of the nucleus as

T(N,Z)=\frac{3}{5}\left[N\epsilon_F(N)+Z\epsilon_F(Z)\right]

, set

N=\frac{A}{2}+\delta

Z=\frac{A}{2}-\delta

with

\delta=\frac{N-Z}{2}

, and expand to second order in

\delta

to show that

T(N,Z) \approx T_{\rm sym} + \frac{1}{3}\epsilon_F\frac{(N-Z)^2}{A},

where

T_{\rm sym}=\frac{3}{5}\epsilon_F A

is the kinetic energy of symmetric matter of the same

A

. (The first-order term in

\delta

vanishes because

N^{5/3}+Z^{5/3}

is symmetric under

\delta\to-\delta

.) ((b)) Taking

\epsilon_F=37\,\mathrm{MeV}

, evaluate the kinetic-asymmetry coefficient

\frac{1}{3}\epsilon_F

numerically, and use it to compute the kinetic-asymmetry contribution to the energy of

^{120}\mathrm{Sn}

(

Z=50

N=70

). ((c)) The fitted asymmetry coefficient of the mass formula is

a_A=23\,\mathrm{MeV}

, giving a total asymmetry energy

a_A(N-Z)^2/A

. What fraction of this total does the kinetic part found in part (b) account for, for

^{120}\mathrm{Sn}