https://dl.dropbox.com/u/67511965/PHY4504/PHY4504-Lectures/LIQUID %20DROP%20MODEL.doc LIQUID DROP MODEL - A non-rotating drop of liquid in the absence of gravitational or other external fields adjusts its shape to minimize its energy. That shape is spherical and it minimizes the positive surface tension energy. - If the liquid is incompressible then the drop's density is constant, independent of radius R and R is given , where n is the number of molecules in the drop. - Let each molecule (except one in or near the surface) be bound in the drop with energy a; this is the energy required to remove the molecule from the inside of the drop and is due to the forces that can exist between molecules, - Typically these forces are negligible at large separations, can become attractive at separations comparable to the molecular size and become strongly repulsive at closer separations. 1 https://dl.dropbox.com/u/67511965/PHY4504/PHY4504-Lectures/Liquid %20drop%20Model%20-%20Edit.doc https://dl.dropbox.com/u/67511965/PHY4504/PHY4504%20Links.doc
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- A non-rotating drop of liquid in the absence of gravitational or other external fields adjusts its
shape to minimize its energy. That shape is spherical and it minimizes the positive surface
tension energy.
- If the liquid is incompressible then the drop's density is constant, independent of radius R and R is given , where n is the number of molecules in the drop.
- Let each molecule (except one in or near the surface) be bound in the drop with energy a;
this is the energy required to remove the molecule from the inside of the drop and is due to
the forces that can exist between molecules,
- Typically these forces are negligible at large separations, can become attractive at separations
comparable to the molecular size and become strongly repulsive at closer separations.
Volume energy. When an assembly of nucleons of the same size is packed together into the
smallest volume, each interior nucleon has a certain number of other nucleons in contact with it.
So, this nuclear energy is proportional to number of particle n in the volume. Equal to
Surface energy. A nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area. Equal to
(T is the surface tension) or
- Therefore the binding energy B of the drop:
where β contains all the constants of the surface term.
In fact, listing in units of ΔE, as each step (which changes proton to neutron) is made, we find the changes require energy in units of ΔE
1, 1, 3, 3, 5, 5, 7, ...
so that the cumulative effect is
1, 2, 5, 8, 13, 18, 25, 32, …… unit ΔE
for N-Z = 2, 4, 6, 8, 10, 12, 14, 16, … ,
Therefore to change from N-Z= 0 to N > Z, with A = N+Z held constant, requires an energy of
.
- This is independent of whether it is N or Z that becomes larger and it means that, if all other things are equal, nuclei with Z= N have less energy and are therefore more strongly bound than a nucleus with Z≠ N.
- Thus we must add a term which reduces the binding energy when Z≠N. Since the energy
levels of a particle in a potential well have a spacing inversely proportional to the well
volume, we can put ΔE A-1.
- Therefore we include a term which reduces the binding energy for nuclei for which Z≠N. This is the asymmetry term:
to be added to the binding energy formula.
- The pairing term. It reflects the fact that it is found experimentally that 2 protons or 2 neutrons
are always more strongly bound than 1 proton and 1 neutron. That is, like nucleons 'pair'.
Before putting the formula for binding energy together we note that there is one refinement that is sometimes made.
The charge on the nucleus is carried in discrete units, one on each proton. The charge on the
proton does not interact with itself (or if the proton constituents do interact, that energy is already
included in the proton mass). It is therefore sensible to replace Z2 appropriate to a continuous
charge distribution by Z(Z-1) which is appropriate to this discreteness of the nuclear charges.
However, we do not do that: the reason is that the apparently best set of coefficients av, etc. has
been determined using the formula with Z2. In addition, the final precision of the formula is
probably not sufficient to allow improvements at this level to be discerned.
So putting all our terms together, we have
The values of the coefficients have to be found by fitting to the binding energy data for medium and heavy nuclei. The light nuclei (A<20) are not included as there is no smooth curve of binding energy against A or Z due to the effects of shell closures.
The fit is not perfect because these effects persist throughout the periodic table and because some nuclei are not spherical.
We have written the whole formula for the nuclear mass (Table 4.1) but as rest mass energy—hence the c2 attached to the real masses. Also shown is a favoured set of values for the coefficients.
Fig. 4.5 shows how the various contribution (Except the pairing term) change with A throughout the periodic table.
Fig 4.5
What is surprising is that this formula is good from A≈20 to the end of the periodic table with a precision beter than 1.5 present on the binding energy. This is shown in the case of the odd A nuclei in Fig. 46.