bei48482_FM

Corrections to the Formula

The binding-energy formula of Eq. (11.13) can be improved by taking into account

two effects that do not fit into the simple liquid-drop model but which make sense in

terms of a model that provides for nuclear energy levels. (We will see in the next sec-

tion how these apparently very different approaches can be reconciled.) One of these

effects occurs when the neutrons in a nucleus outnumber the protons, which means

that higher energy levels have to be occupied than would be the case if Nand Zwere

equal.

Let us suppose that the uppermost neutron and proton energy levels, which the

exclusion principle limits to two particles each, have the same spacing , as in

Fig. 11.16. In order to produce a neutron excess of, say, NZ8 without chang-

ing A, ^12 (NZ)4 neutrons would have to replace protons in an original nucleus

in which NZ. The new neutrons would occupy levels higher in energy by

2  4 2 than those of the protons they replace. In the general case of ^12 (NZ)

new neutrons, each must be raised in energy by 2 ^1 (NZ)2. The total work

needed is

E(number of new neutrons)

 (NZ) (NZ)  (NZ)^2

Because NAZ, (NZ)^2 (A 2 Z)^2 , and

E (A 2 Z)^2 (11.15)

As it happens, the greater the number of nucleons in a nucleus, the smaller is the

energy level spacing , with proportional to 1A. This means that the asymmetry

energyEadue to the difference between Nand Zcan be expressed as

Asymmetry energy EaEa 4 (11.16)

The asymmetry energy is negative because it reduces the binding energy of the

nucleus.

(A 2 Z)^2



A





8





8





2

1



2

1



2

energy increase



new neutron

406 Chapter Eleven

Figure 11.16In order to replace 4 protons in a nucleus with NZby 4 neutrons, the work

(4)(42) must be done. The resulting nucleus has 8 more neutrons than protons.

Energy

Neutron

Proton

e

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