bei48482_FM

Nuclear Structure 405

Since there are Z(Z1)2 pairs of protons,

Ec V    

av

(11.11)

where (1r)avis the value of 1raveraged over all proton pairs. If the protons are

uniformly distributed throughout a nucleus of radius R, (1r)avis proportional to 1R

and hence to 1A^1 ^3 , so that

Coulomb energy Eca 3 (11.12)

The coulomb energy is negative because it arises from an effect that opposes nuclear

stability.

This is as far as the liquid-drop model itself can go. Let us now see how the result

compares with reality.

The total binding energy Ebof a nucleus ought to be the sum of its volume, surface,

and coulomb energies:

EbEEsEca 1 Aa 2 A^2 ^3 a 3 (11.13)

The binding energy per nucleon is therefore

a 1 a 3 (11.14)

Each of the terms of Eq. (11.14) is plotted in Fig. 11.15 versus A, together with their

sum EbA. The coefficients were chosen to make the EbAcurve resemble as closely

as possible the empirical binding energy per nucleon curve of Fig. 11.12. The fact that

the theoretical curve can be made to agree so well with the empirical one means that

the analogy between a nucleus and a liquid drop has at least some validity.

Z(Z1)



A^4 ^3

a 2



A^1 ^3

Eb



A

Z(Z1)



A^1 ^3

Z(Z1)



A^1 ^3

1



r

Z(Z1)e^2



8  0

Z(Z1)



2

50 100 150 200 250

15

10

5

0

• 5


• 10

A

Eb

/A

, MeV

Volume energy

Total energy

Surface Coulomb energy

energy

Figure 11.15The binding energy per nucleon is the sum of the volume, surface, and coulomb energies.

bei48482_ch11.qxd 1/23/02 3:14 AM Page 405 RKAUL-9 RKAUL-9:Desktop Folder: