about liquid drop model in nuclear physics

2. LIQUID DROP MODEL
 The liquid-drop model was first proposed by George Gamow and further developed by Niels Bohr and John
Archibald Wheeler.
 It treats the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force
(a residual effect of the strong force), there is a similarity to the structure of a spherical liquid drop. The
liquid-drop model accounts for the spherical shape of most nuclei and makes a rough prediction of binding
energy.
 In nuclear physics, the semi-empirical mass formula (SEMF) is used to approximate the mass of an atomic
nucleus from its number of protons and neutrons. As the name suggests, it is based partly on theory and partly
on empirical measurements. The formula represents the liquid-drop model proposed by George Gamow ,
which can account for most of the terms in the formula and gives rough estimates for the values of the
coefficients.

3. LIQUID DROP MODEL
The idea that the molecules in the drop of liquid corresponding to the
nucleons in the nucleus is confirmed due to the following similarities-
1.The nuclear force is analogous to the surface tension force of a
liquid.
2.The nucleons behave in a manner similar to that of molecules in a
liquid.
3.The density of the nuclear matter is almost independent of A,
showing resemblance to liquid drop.
4.The constant binding energy per nucleon is analogous to the latent
heat of vaporization.
5.The disintegration of nuclei by the emission of particles is
analogous to the evaporation of molecules from the surface of the
liquid.
6.The energy of nuclei corresponds to internal thermal vibrations of
drop molecules.
7.The absorption of bombarding particles by a nucleus corresponds to
the condensation or drops.

4. LIQUID DROP MODEL
We start by assuming that the energy associated with each nucleon-
nucleon bond has some value U. This energy is actually negative
since attractive forces are involved, but is usually written as
positive because binding energy is considered a positive quantity
for convenience.
Because each bond energy U is shared by two nucleons, each has a
binding energy of 1/2 U. When an assembly of spheres of the same
size is packed together into the smallest volume, as we suppose is
the case of nucleons within a nucleus, each interior sphere has 12
other spheres in contact with it . Hence each interior nucleon in a
nucleus has a binding energy of (12)(1/2 U) or 6 U. If all A
nucleons in a nucleus were in its interior, the total binding energy
of the nucleus would be

5. LIQUID DROP MODEL
The binding energy EB can be expressed as the sum of a number of terms given below:
1. Volume Energy
2. Surface Energy
3. Coulomb Energy
4. Asymmetry Energy
5. Pairing Energy

6. LIQUID DROP MODEL

7. LIQUID DROP MODEL

8. LIQUID DROP MODEL

9. LIQUID DROP MODEL
The total binding energy EB of a nucleus ought to be the sum of
its volume, surface, and coulomb energies:
The binding energy per nucleon is therefore
Each of the terms of is plotted in Fig(a) versus A,
together with their sum EB/A. The coefficients were
chosen to make the EB/A curve resemble as closely
as possible the empirical binding energy per nucleon
curve of Fig(b).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.
Fig(a)
Fig(b)

10. LIQUID DROP MODEL
Correction to the formula
Correction 1:
When the neutrons in a nucleus outnumber the protons, which means
that higher energy levels to be occupied than would be the case if N and
Z were equal.
The asymmetry energy Ea due to the difference between N and Z can be
expressed as The asymmetry energy is negative
because it reduces the binding
energy of the nucleus.

11. LIQUID DROP MODEL
Correction 2:
The correction term arises from the tendency of proton pairs and neutron pairs to occur .Even-even
nuclei are the most stable and hence have higher binding energies . Thus such nuclei as
appear as peaks on the empirical curve of binding energy per nucleon. At the other extreme, odd-odd
nuclei have both unpaired protons and neutrons and have relatively low binding energies.
 The pairing energy Ep is positive for even-even nuclei.
 The pairing energy Ep zero for odd-even and even-odd nuclei
 The pairing energy Ep negative for odd-odd nuclei, and seems to vary with A as A-3/4.
Hence , pairing energy is given by

12. The final expression for the binding energy of a nucleus of atomic
number Z and mass number A is
LIQUID DROP MODEL
Value of different constants in MeV in Semi-Empirical mass
formula :
 av = 14.1 MeV
 as = 13.0 MeV
 ac = 0.595 MeV
 aa = 19.0 MeV
 ap = 33.5 MeV

13. LIQUID DROP MODEL
Applications of the Semi-Empirical mass formula :
 Alpha decay
 The mass parabolas and prediction of stability against beta activity
 β-disintegration energy of the mirror nuclei