NUCLEAR MODELS
In the absence of a detailed theory of nuclear structure,
attempts have been made to correlate nuclear data in terms of rough
pictures or "models" of the nucleus.
Several models have been proposed, each of which is useful in
a limited way. Each is used to explain some nuclear phenomena but
fails when applied to data outside its range.
The models we shall discuss are : the "liquid drop" model
which accounts for the nuclear binding energy ; the "shell model"
which accounts for the existence of stable isotopes and describes
their energy states and their angular momenta ; and also the
"collective model" including some rotational and vibrational states.
1. LIQUID - DROP MODEL
1-1 Liquid-drop Model
A nucleus may be considered to be analogous to a drop of
incompressible fluid of very high density —1014 g/ cm3). This is
encouraged by the fact that some of the properties of nuclear forces
(saturation and short-range) are analogous to the properties of
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forces which hold a liquid drop together. The essential assumptions
are
(i) the nucleus consists of incompressible matter so that R ∝A1/3
(ii) the nuclear force is the same for every nucleon and does not
depend on whether it is a neutron or a proton.
(iii) the nuclear force saturates
This idea has been used together with other classical ideas such as
electrostatic repulsion and surface tension to set a semi-empirical
formula for the mass or binding energy of a nucleus in its ground
state.
1-2 Semi-empirical Mass Formula
The formula has been developed by considering the different
factors which affect the binding energy, and weighting these factors
with constants derived from theory when possible and from
experimental data where theory cannot help.
Contributions to the total binding energy of the nucleus :
(l) Volume energy: The main contribution to Bt comes from a term
proportional to the mass no. A, since the volume of the nucleus is
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also proportional to A , this term is regarded as a "volume energy".
𝑩𝒕 = 𝒂𝒗 𝑨
(7-1)
(2) Surface energy: Eq. (7-1) is the binding energy for an
"infinitely" large nucleus. Actual nuclei are finite and are usually
spherical in shape; hence nucleons on the surface are not attracted
as much as estimated. A term proportional to the surface area must
be subtracted from the infinite nucleus expression.
(3) Coulomb energy between the protons tends to lower Bt and it is
given as a negative term. It can be shown that the total Coulomb
energy of a nucleus of charge Z is given by .
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(4) Asymmetry energy: The greatest Bt occurs for symmetric nuclei
where Z= N. The asymmetry energy is the difference between the Bt
of a nucleus A (Z,N) and the Bt of the symmetric isobar with
Z=N=1/2A.
To find the difference in b-e . Let us start with the identical energy
states of neutrons and protons in the symmetric nucleus. Now,
transfer some n protons to neutrons in order to reach the
asymmetric case. Now,
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(5) Pairing energy: Experimental results showed that nuclei with
even no. of protons and neutrons (even-even nuclei) are the most
stable and have greatest values of Bt . The "odd-odd" nuclei are
least stable and have the least values of Bt, while the "odd-A" nuclei
are intermediate. A correction term "' is added to Bt to account for
these differences.
+ is added to the Bt in case of even-even nuclei
- is added to the Bt in case of odd-odd
 is considered zero for odd-A nuclei.
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(6) Shell effects: An extra tem  is added to Bt expression to show
the effect of the shell structure on Bt .  has a positive value when N
or Z approaches a "magic number", i.e. approaches a closed shell
configurations. (Magic numbers are 2, 8, 20, 28, 50, 126) .
The total binding energy of a nucleus (Bt) can now take the form
The constants of the semi-empirical mass formula can be
determined by comparison with available data. The "fit" is never
perfect, and hence several sets of coefficients have been used- One
such set is (in MeV)
The contributions of the various terms to the average b.e. per
nucleon, Bave is shown in the figure and it fits with the experimental
curve for Bave .
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7-3 Abundance Systematics of Stable Nuclides
The nuclides found on the earth are either stable or are
radioactive with half-lives longer than 5×109 years (the age of earth)
. The figure below presents an N, Z plot for the known stable
nuclides
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For light nuclides, the average "line of stability" clusters around
N=Z; for heavier ones, it deviates from this because the increasing
effect of the Coulomb repulsion between protons.
We notice that Particularly high stability and high
abundance with respect to neighboring nuclides is associated with
nuclides for which N or Z is equal to 2, 8, 20, 28, 50, 82 and 126.
These "magic numbers" reflect the effects of closed shells (like
closing of electronic shells in atoms)
7-4 Mass Parabolas
7-5:Closing Remarks
(l) Although we have applied the liquid-drop model only to ground
states of nuclei, it can also be used for excited states. These would be
produced by oscillations of the nuclear "drop" or by ripples
traveling over its surface. This idea is particularly useful in
explaining nuclear fission.
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(2) The liquid-drop model stresses cooperative effects between many
nucleons and is the basic idea for the "collective" models.
7-6. SHELL MODEL
An Impressive amount of experimental material has been
accumulated which show periodic variations in many nuclear
properties (similar to the regularities in the chemical and physical
properties of atoms as shown in the periodic table of the elements).
These properties showed marked discontinuities which point to shell
closures at certain even values of proton and neutron numbers.
These numbers, called the "magic numbers" are: 2, 8, 20, 28, 50, 82
and 126
7-7 Experimental Basis of the Shell Model
(l) The abundance data of stable isotones (nuclides of the same N
and different Z) show that the number of stable isotones is
particularly high when N is a magic number
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(2) The separation energy for a neutron Sn is very high when N is a
magic number (because it is very hard to separate a neutron from a
closed shell). On the other hand, Sn becomes very low when
N=(magic no. + l).
(3) Magic nuclei, being more tightly bound, require more energy to
be excited than non-magic nuclei.
(4) Magic N nuclei have smaller fast-neutron capture sections. (The
cross section is proportional to the probability of a nuclear
reaction). This can be explained by the saturation of the neutron
closed shells.
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7-8 Basic Assumptions of Shell Model
(1) In spite of the strong interactions between nucleons, each
nucleon is assumed to move in its orbit Independent of that of other
nucleons. The orbit is determined by the potential energy function
V(r) which represents the average effect of all interactions with
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other nucleons. If all inter-nucleon couplings (called residual
interactions) are ignored, we call the model the "single-particle shell
model"
(2) Neutrons and protons are arranged in separate shells and each
shell has a certain maximum number of neutrons or protons.
(3) When a shell is completely filled, the resulting configuration is
particularly stable and has a lower value of energy.
(4) Pauli exclusion principle is applied to both protons and neutrons.
It excludes two protons or two neutrons from occupying the same
quantum state, i.e., two identical nucleons cannot have the same set
of quantum numbers.
7-9 Theoretical Solutions
The aim of theoretical studies is to solve the Schrodinger
equation for a particle moving in a spherically symmetric central
field of force- In this case, we can show that the general wave
function can be separated in spherical coordinates r,  and
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13
14
15
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7-9C Spin - Orbit Coupling (𝓵 − 𝑺 coupling)
The successful trial was done independently by Mayer and Haxel in
1949. They proposed a non-central component to the nuclear force.
It is the strong interaction existing between the orbital angular
momentum 𝓵ℏ and the intrinsic spin angular momentum sℏ, the two
angular momenta couple to form the total angular momentum jℏ. In
terms of quantum numbers, we get that
J= 𝓵 ±
𝟏
𝟐
(6-26)
Due to this strong interaction, a different energy is associated with
each value of j , giving rise to the splitting of the levels due to spinorbit coupling.
Remarks.
(i) The term with the higher j value is more stable so it has a
lower energy value.
(ii) The energy separation of the two levels is roughly
proportional to the value of 𝓵 Sometimes the levels with
high j values are lowered very much so that they appear
with levels of lower N value (lower shell)
(iii) Each state is occupied by a number of nucleons equal to
(2j+1)
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(iv) In spectroscopic notation, the j value is placed as a subscript
to the (n, 𝓵) symbol. For example, for the 1p shell, the
splitting produces 1p1/2 and 1p3/2 levels .
We can see from the figure that when the value of the spin-orbit
coupling is properly adjusted; the major shell breaks occur at the
experimentally determined magic numbers.
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7-10 Applications of Shell Model
(l) Assuming that the angular momentum of odd-A nuclei is
determined by the odd-nucleon, a remarkable agreement is obtained
between the ground state spins (and parities) of these nuclei and the
predictions of the spin-orbit coupling
(2) The long-lived isomers (having half-lives greater than one
second) are found to exist at mass numbers just below magic
numbers, and when the shell is completely filled the isomer states
disappear. These long-lived states are caused by large angular
momentum differences and small energy differences which make the
transitions highly forbidden (having very small probability thus
very long half-lives).
(3) Experimental magnetic moment values and electric quadrupole
measurements are in good agreement with the calculations based on
the shell model. Some large quadrupole moment values indicate that
the nucleus is not spherical but rather spheroidal in shape, which
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led to developing the ideas of the "collective" motion of nucleons.
7-11 Angular Momenta of G.S.
For even-even nuclei:
(1) For even-even nuclei: It is found experimentally that all eveneven nuclei have zero g.s., spins. -It follows that in the g.s. of
any nucleus, the net angular momentum associated with an
even N or Z is equal to zero.
(2) For odd-A nuclei, the angular momentum of the nucleus is
determined solely by that of the last odd nucleon.
Example: For 7Li3 nucleus, the odd proton lies in the 1p3/2
state- The angular momentum of Li nucleus is I=3/2
(3) For odd-odd nuclei, the angular momentum of the odd
proton, Ip , couples with that of the odd neutron, In , where
I= Ip + In
(Vector addition)
(7-27)
In terms of quantum numbers, we have
I= (Ip+In) , (Ip+In-1),
………..|𝑰𝒑 + 𝑰𝒏 |
(7-28)
Ground states of nuclei are usually formed when the intrinsic
spins of the proton and the neutron are parallel -
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Example: In the case of the deuteron, Ip = ½ and In = ½ the total
ang. momentum I=0 or I=1
The g.s. of deuteron has I = 1 , because the g.s. favors the parallel
spins (and this is confirmed experimentally)
7-12 Parities of G.S.
The parity of the wave function describing a nuclear state is
determined by the orbital motion of the nucleon. Since the angular
part of the wave function is proportional to the associated Legendre
polynomial of the order 𝓵 in cos, 𝑷𝓵 (cos ) , the parity is given  is
given by
𝝅 = (−𝟏)𝓵
(7-29)
(l) For even-even nuclei , all the g.s. have I=0 , thus 𝓵 = 0 and they
all have even parities ( + l)
(2) For odd-A nuclei, the parity for the nucleus is determined by the
parity of the last odd nucleon.
Example : For 15N7 nucleus, the odd proton lies in the 1pl/2 state, it
has an odd parity- thus 𝓵 = 𝟏 and  =-1, it has an odd parity
(3) For odd-odd nuclei , the parity of g.s. of the nucleus is the
product of the parities of the odd proton (p) and that of the
odd neutron ( n) , thus
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 (total) =p +  n
(7-30)
Example: For 6Li3 , the odd proton and the odd neutron both lie in
the 1p3/2 state . they both have 𝓵 = 𝟏 thus p = -1 and also n =-1 .
The parity of g.s. of 6Li is even (l)
7-13 Closing Remarks
(l) Although the shell model explains successfully many aspects of
the nuclear structure yet it is not sufficient to explain other aspects.
There is mounting evidence that there are other residual forces
between nucleons, forces other than that already taken into account
in the shell model potential.
(2) One of the residual forces is the nucleon-nucleon Interaction that
favors the paring of nucleons with opposing angular momenta. An
attractive force must be added to the single nucleon spin-orbit
interaction, which gives rise to the pairing energy.
The magnitude of the pairing energy increases with the 𝓵 value of
the pair. For this reason, the high-spin states like h11/2 and i13/2
predicted by the model are not found in g.s. of odd-A nuclei. For
example, the energy of six nucleons in 2d5/2 plus one in 1hl1/2 is
higher than the energy of five nucleons in 2d5/2 plus two paired in
h11/2 , the latter system is lower in energy and is favored in nature.
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iii. COLLECTIVE MODEL
For heavy nuclei, many predictions of the single-particle shell model
do not agree quantitatively with experiment. The discrepancies are
particularly severed for magnetic dipole moments. Also, the shell
model predicts vanishingly small quadrupole moments for closed
shells, and quadrupole moments of opposite sign for neighboring
nuclei with atomic numbers 𝐙 ± 𝟏.although this agrees qualitatively
with experiment, the measured values of quadrupole moments are
very different from the predictions. In fact some heavy nuclei
appear to have large permanent electric quadrupole moments,
suggesting a nonsphericity in the shape of these nuclei. This is
certainly not consistent with the assumptions of the shell model,
where rotational symmetry plays a crucial role.
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In a revival of the liquid drop model, Aage Bohr noted that
many properties of heavy nuclei could be attributed to a surface
motion of the nuclear liquid drop. Furthermore, James Rainwater
showed that excellent agreement between the expected and
measured values of magnetic dipole and electric quadrupole
moments could be obtained under the assumption that the liquid
drop had an aspherical shape. These successes presented some- what
of a dilemma because the liquid drop model and the single-particle
shell model had fundamentally opposite viewpoints about the nature
of nuclear structure. Individual particle characteristics, such as
intrinsic spin and orbital angular momentum, play no role in a
liquid drop picture, where collective motion that involves the entire
nucleus has prime importance. On the other hand, individual
nucleon properties, especially of the valence nucleons, are crucial to
the success of the independent-particle shell model. The shell model
had yielded too many important nuclear features to be abandoned
outright, and reconciliation between the two extreme views was
needed.
The reconciliation was brought about by Aage Bohr, Ben
Mottelson and James Rainwater who proposed a collective model
for the nucleus that provided many features that were not present in
either the shell or the liquid drop model. In what follows, we
describe this model only qualitatively. Its basic assumption is that
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1. a nucleus consists of a hard core of nucleons in the filled shells,
and outer valence nucleons that behave like the surface
molecules in a liquid drop.
2. The surface motion (rotation) of the valence nucleons
introduces a nonsphericity in the central core, which in turn
affects the quantum states of the valence nucleons.
In other words, one can think of the surface motion as a
perturbation that causes the quantum states of the valence
nucleons to change from the unperturbed states of the shell
model. This adjustment accounts for the difference in predictions
for dipole and quadrupole moments from those given by the shell
model. Physically, one can view the collective model as a shell
model with a potential that is not spherically symmetric.
Spherically symmetric nuclei are, of course, insensitive to
rotations, and consequently rotational motion cannot produce
additional (rotational) energy levels in such nuclei. Aspherical
nuclei, on the other hand, can have additional energy levels
because of the presence of rotational and vibrational degrees of
freedom. These types of effects modify the predictions of the
simple shell model. In particular, large nonsphericity in nuclei
can provide large permanent dipole and quadrupole moments.
Mathematically, these ideas can be incorporated as follows. For
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simplicity, we assume the nucleus to be an ellipsoid defined by the
form
Where a and b are parameters related to the deformation from a
spherical shape of radius R and fixed volume 4/3 R3. The mean
potential for nuclear motion can then be chosen as
Needless to say, more realistic calculations in the collective model
provide even better descriptions of nuclear properties, but they also
become far more complicated.
One of the important predictions of the collective model is the
existence of rotational and vibrational levels in a nucleus. These
levels can be derived much the same way as is done for the case of
molecules. Thus, we can choose the Hamiltonian for rotations to be
with eigenvalues
𝓵(𝓵+𝟏)
𝟐𝐈
ℏ𝟐 where the effective moment of inertia I is
a function of the nuclear shape. If there is rotation about an axis
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perpendicular to the symmetry axis of the ellipsoid, it can then be
shown that the angular momentum of the rotational levels can only
be even. Thus, we see that rotational and vibrational levels in a
nucleus are predicted with specific values of angular momentum
and parity. Such excitations have indeed been found through the
observation of photon quadrupole transitions (𝚫𝓵 = 𝟐) between
levels.
Finally, the collective model accommodates quite naturally
the decrease, with increasing A, of the spacing between the first
excited state and the ground level in even-even nuclei, as well as the
fact that the spacing is largest for nuclei with closed shells. The first
follows simply because the moment of inertia grows with A, which
decreases the energy eigenvalue of the first excited rotational state.
The latter is due to the fact that a nucleus with a closed shell should
not have a rotational level because such a nucleus would tend to be
spherical. On the other hand, such a nucleus can have vibrational
excitations. However, vibrational excitations involve the entire core
and not just the surface. The core being much more massive implies
that the energy level for vibration will lie far higher, and the spacing
between the ground state and the first excited state will be much
greater.
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