PHP480: NUCLEAR PHYSICS
Course In-charge: Dr. Baskaran Rangasamy
Study Material for Week- 3 (06.04.2020 to 10.04.2020)
The Shell Model
The shell model of the nucleus assumes that the energy structure (energy levels of the
nucleons) of the nucleus is similar to that of an electron shell in an atom. According to this
model, the protons and neutrons are grouped in shells in the nucleus, similar to extra-nuclear
electrons in various shells outside the nucleus. The shells are regarded as "filled" when they
contain a specific number of protons or neutrons or both: The number of nucleons in each
shell is limited by the Pauli exclusion principle. The shell model is sometimes referred to as
the independent particle model because it assumes that each nucleon moves independently of
all the other nucleons and is acted on by an average nuclear field produced by the action of all
the other nucleons.
Evidence for shell model
It is known that a nucleus is stable if it has a certain definite number of either protons
or neutrons. These numbers are known as magic numbers. The magic numbers are 2, 8, 20,
50, 82 and 126. Thus nuclei containing 2, 8, 20, 50, 82 and 126 nucleons of the same kind
form some sort of closed nuclear shell structures. The main points in favour of this inference
are:
1) The inert gases with closed electron shells exhibit a high degree of chemical stability.
Similarly, nuclides whose nuclei contain a magic number of nucleons of the same kind
exhibit more than average stability.
2) Helium 4 (Z = 2, N = 2) and oxygen 16 (Z = 8, N = 8) are particularly stable as evidenced
from the binding energy curve. Thus the numbers 2 and 8 express stability.
3) Relative abundance: Isotopes of elements having an isotopic abundance greater than 60%
belong to the magic number category.
4) Stable isotopes: Tin (50Sn) has ten stable isotopes, while calcium (20Ca40) has six stable
isotopes. So elements with Z = 50, 20 are more than usually stable.
5) Stable end products of three radioactive series: The three main radioactive series (viz.,
the uranium series, actinium series and thorium series) decay to 82Pb208 with Z = 82 and
N = 126. Thus lead 82Pb208 is the most stable isotope. This again shows that the numbers
82 and 126 indicate stability.
6) Neutron capture cross section: Nuclei having a number of neutrons equal to the magic
number cannot capture a neutron because the shells are closed and they cannot contain an
extra neutron.
7) Some isotopes are spontaneous neutron emitters when excited above the nucleon binding
energy by a preceding β-decay. These are 8O17, 36Kr87 and 54Xe137 for which N= 9, 51 and
83 which can be written as 8+1, 50+1, and 82+1. If we interpret this loosely bound
neutron as a valency neutron, the neutron numbers 8, 50, 82 represent greater stability than
other neutron numbers.
If we interpret this loosely bound neutron as a valency neutron, the neutron numbers
8, 50, 82 represent greater stability than other neutron numbers. For example the nuclide
87
16Kr with N = 51 is a neutron emitter, N = 50 is a stable configuration or a closed nuclear
shell. Thus we find that the numbers 2, 8, 20, 50, 82, 126 for either Z or N indicate greater
nuclear stability. These are called magic numbers and the nuclei are called magic nuclei.
It is apparent from the above conclusions that nuclear behaviour is often determined
by the excess or deficiency of nucleons with respect to closed shells of nucleons
corresponding to the magic numbers. It was, therefore, suggested that nucleons revolve
inside the nucleus just as electrons revolve outside in specific permitted orbits.
Magic Number Predictions
It has been found that the magic numbers (2, 8, 20, 50, 82, 126) are specially favoured
when changes of nuclear property are studied. In general it has been observed that plots of
many nuclear properties against Z or N show characteristic peaks or points of inflection at
these magic numbers.
The magic numbers can be theoretically predicted using the possible L-S and j-j
coupling. Let us consider the orbital angular momentum quantum number of the nucleons so
that i = 0, 1, 2, 3 could possibly depict shells closing at 2, 8, 20, 50, 82, 126. Taking the case
of orbital coupling only, it is seen that the shells might be closed at nucleon numbers given by
2 (2l+1) protons or neutrons for l= 0, 1, 2, 3 etc. But in this way the lower magic numbers can
be obtained. The higher magic numbers can however be predicted using a model of the
nucleus in which each nucleon has an angular momentum (1/2.ℏ) due to spin and an orbital
angular momentum of l.ℏ. These combine to give the total angular momentum,
j = l ± 1/2 in terms of ℏ.
According to Pauli's Exclusion Principle, (2j + 1) nucleons can have an angular
momentum j in a given nucleus. Making these assumptions, it is possible to proceed through
the elements filling up successive energy levels with nucleons and predicting the magic
numbers which correspond to completed energy levels or shells within the nucleus. This is
closely analogous to the way in which the electronic shells of the atom are built up. Each
shell is limited to a certain maximum number of protons and neutrons. When a shell is filled,
the resulting configuration is particularly stable and has an unusually low energy.
Calculation of Potential Field
Since the nuclear forces are as yet not fully known, we cannot calculate the potential
field, but it is reasonable to assume that it is fairly constant within the nucleus and changes
rapidly near the edges. Various potential shapes have been suggested to derive the magic
numbers theoretically of which the simplest ones are :
(a) Infinite rectangular-well potential of the form
V(r) = −𝑉 for r < r0
V(r) = −∞ for r < r0
(b) Infinite harmonic oscillator potential (which is most suited to light nuclei) of the
form
V(r) = −𝑉 [1 −
)
=
𝑚𝜔 (𝑟 − 𝑟 )
where ω is the angular frequency of the harmonic oscillator of mass m.
In the one dimensional case, it is well known that the energy levels are given by
𝐸 = (𝑛 + )ℏ𝜔
and in the general three-dimensional
dimensional case by
𝐸
= (𝑛 + 𝑛 + 𝑛 + )ℏ𝜔
𝐸 = (𝑁 + )ℏ𝜔
where n1, n2, n3 are the integers specifying the wave functions and N = n1 + n2 + n3 (≥ 0) is
the oscillator quantum number. When the angular dependence of each wave function is
examined, it is found that for each N value there is a degenerate group of levels with
wi different
values of l such that l ≤ N and even N correspond to even l and odd N correspond to odd l.
Thus for N = 2 both s and d states occur with the same energy.
Fig.1: Energy levels of nucleons (a) in an infinite spherical well and (b) in a para
parabolic potential well
The number of nucleons which may be accommodated in the levels described by the
oscillator N is found to be (N + 1)
1). (N + 2). The levels are shown in the Fig 1. It is seen that
the order of the levels is
ls, 1p, ld, 2s; lf, 2p, …
and the shells are complete at numbers
2, 8, 20, 40, 70, 112.
This shows that all the magic numbers are not predicted but only the first three lower magic
numbers are available.
Main Assumptions of the Shell Model
1) The protons and neutrons move in two separate systems of orbits round the centre of mass
of all the nucleons.
2) The extra-nuclear electrons revolve in the Coulomb field of a relatively distant heavy
nucleus. But the nucleons move in orbits around a common centre of gravity of all the
constituents of the nucleus.
3) Each nucleon shell has a specific maximum capacity. When the shells are filled to
capacity, they give rise to particular numbers (the magic numbers) characteristic of
unusual stability.
Predictions of the Shell Model
The shell model is able to account for several nuclear phenomena in addition to magic
numbers.
1) It is observed that even-even nuclei are, in general, more stable than odd-odd nuclei. This
is obvious from the shell model. According to Pauli's principle, a single energy sublevel
can have a maximum of two nucleons (one with spin up and other with spin down).
Therefore, in an even- even nucleus only completed sublevels are present which means
greater stability. On the other hand, an odd-odd nucleus contains incomplete sublevels for
both kinds of nucleon which means lesser stability.
2) The shell model is able to predict the total angular momenta of nuclei. In even-even
nuclei, all the protons and neutrons should pair off so as to cancel out one another's spin
and orbital angular momenta. Thus even-even nuclei ought to have zero nuclear angular
momenta, as observed. In even-odd and odd-even nuclei, the half-integral spin of the
single "extra" nucleon should be combined with the integral angular momentum of the rest
of nucleus for a half-integral total angular momentum.
3) Odd-odd nuclei each have an extra neutron and an extra proton whose half-integral spins
should yield integral total angular momenta. Both these predictions are experimentally
confirmed.
Nuclear Energy Level Scheme and Explanation of Magic Numbers
(Spin-Orbit coupling model)
To account for the observed magic numbers, Mayer and Jensen postulated a strong
nuclear spin-orbit interaction. The magnitude of the spin-orbit interaction is such that the
consequent splitting of energy levels into sublevels is many times larger than the analogous
splitting of atomic energy levels. The nuclear spin-orbit splitting of a single-nucleon energy
level is assumed to be large and also inverted (Fig.2). We ascribe this behaviour to a nuclear
interaction of the form,
VSL (nucleus) = −
𝑆. 𝐿
The minus sign accomplishes the required inversion of the split levels.
j=l−
nl
j = l+
Fig.2: Spin-orbit splitting of a single nucleon energy level
The constant 𝑎
produces the desired amount of energy splitting. The central-field
function V (r) appears along with the orbital and spin angular momenta of the nucleon. The
exact form of the potential-energy function is not critical, provided that it more or less
resembles a square well.
The shell theory assumes that LS coupling holds only for the very lightest nuclei, in
which the l values are necessarily small in their normal configurations. In this scheme, the
intrinsic spin angular momenta Si of the particles concerned are coupled together into a total
spin momentum S. The orbital angular momenta Li are separately coupled together into a total
orbital momentum L. Then S and L are coupled to form a total angular momentum J of
magnitude 𝐽(𝐽 + 1)ℏ.
After a transition region in which an intermediate coupling scheme holds, the heavier
nuclei exhibit jj coupling. In this case, the Si and Li of each particle are first coupled to form a
Ji for that particle of magnitude 𝐽(𝐽 + 1)ℏ. The various Ji then couple together to form the
total angular momentum J. The jj coupling scheme holds for the great majority of nuclei.
In this spin-orbit coupling model, the main assumptions are,
1) There is strong spin-orbit coupling in nuclei (i.e. there is large energy dependence on the
relative orientation of spin and orbital angular moments).
2) The spin-orbit doublets are inverted i.e. in contrast to electronic energy levels, the levels of
higher total angular momentum j = l+1/2 have less energy than the levels corresponding
to j = l - 1/2.
Thus the force acting on a nucleon in a nucleus should include a no-central
component. This non-central force arises due to the interaction between the orbital angular
momentum (l) and the spin momentum (S) and this spin-orbit force causes the splitting of the
energy levels giving them a different periodicity. This theory uses a potential intermediate in
shape between the oscillator and the square-well. The energy levels associated with this
intermediate potential are shown in Fig.3.
The first three magic numbers 2, 8, 20 are easily obtained. By the time we reach N =
4, the potential field has changed and becomes more nearly a rectangular-well shape. This has
the effect of increasing the coupling energy of the highest j levels so much so that they
become more closely associated with the next lowest levels. Thus the level at N = 5, j = 9/2
containing 90 nucleons, becomes associated with n = 4 level giving another magic number at
n = 50. Similarly at 82 and 126, j = 11/2 is associated with N = 5 and j = 13/2 is associated
with N = 6. Each change of this type corresponds to depressing the level to a lower energy
level.
Fig.3:: Relation between Shell model and M
Magic numbers
Fig.3 shows the nucleon energy levels according to the shell model. The levels are
designated by a prefix equal to the total quantum number n,, a letter that indicates l for each
particle in that level, and a subscript equal to j. The spin-orbit
orbit interaction splits each state of
given j into 2j+1 substates. The accumulated population of nucleons corresponds to a magic
number at every one of the larger energy gaps. Hence shells are filled when there are 2, 8, 20,
28, 50, 82 and 126 neutrons or protons in a nucleus.
Magnetic Dipole Moments and the Shell Model
Since nuclei with an odd number of protons and/or neutrons have intrinsic spin they
also in general possess a magnetic dipole moment. From the theory of shell model, it is clear
that in an odd nucleus there wi
will be one nucleon left unpaired. The total angular momentum J
of the nucleus is equal to the angular momentum j of the last unpaired
ed nucleon. It is therefore
expected that the magnetic dipole moment of the nucleus is solely due to the odd particles
alone. Let us find whether the experimental and theoretical results agree or not.
If the last unpaired nucleon is an S- state, the magnetic
ic dipole moment will be equal
to the magnetic dipole moment associated with the spin of the unpaired nucleon i.e. 2.79275
nm, if it is a proton and 1.9135
9135 nm if it is a neutron. The orbital angular momentum vector L
1/2
having numerical value [l(l+1)]
+1)]1/2 and the spin vector S with numerical value is [s(s+1)]
[
combine to form the total angular momentum vector J with numerical value [j(j+1)]1/2 all in
units of ℏ. The unit of magnetic dipole moment for a nucleus is the “nuclear magneton”
defined as,
ℏ
𝜇 =
which is analogous to the Bohr magneton but with the electron mass replaced by the proton
mass. It is defined such that the magnetic moment due to a proton with orbital angular
momentum l is µN l.
Experimentally it is found that the magnetic moment of the proton (due to its spin) is,
𝜇 = 2.79𝜇 = 5.58 𝜇 𝑠, (𝑠 =
)
And that of the neutron is,
𝜇 = −1.91𝜇 = −3.82 𝜇 𝑠, (𝑠 =
)
If we apply a magnetic field in the z-direction to a nucleus then the unpaired proton
with orbital angular momentum l, spin s and total angular momentum j will give a
contribution to the z− component of the magnetic moment.
𝜇 = (5.58 𝑠 + 𝑙 )𝜇
As in the case of the Zeeman effect, the vector model may be used to express this as,
.
𝜇 =
Using
.
.
𝑗 𝜇
< 𝑗 > = 𝑗 (𝑗 + 1)ℏ
< 𝑠. 𝑗 > =
(< 𝑗 > +< 𝑠 > −< 𝑙 >)
ℏ
=
< 𝑙. 𝑗 > =
(𝑗(𝑗 + 1) + 𝑠(𝑠 + 1) − 𝑙(𝑙 + 1))
(< 𝑗 > +< 𝑙 > −< 𝑠 >)
ℏ
=
(𝑗(𝑗 + 1) + 𝑙(𝑙 + 1) − 𝑠(𝑠 + 1))
We end up with expression for the contribution to the magnetic moment
𝜇=
(
.
)
(
)
(
)
(
( (
)
(
)
(
))
)
𝑗𝜇
and for a neutron with orbital angular momentum 𝑙 and total angular momentum 𝑗 we get
(not contribution from the orbital angular momentum because the neutron is uncharged)
𝜇= −
(
.
(
)
𝑗𝜇
)
Thus, for example if we consider the nuclide 𝐿𝑖 for which there is an unpaired proton in the
2p3/2 state (l = 1, j = 3/2) then the estimate of the magnetic moment is,
𝜇=
.
×
×
×
× ×
×
×
×
= 3.79𝜇
The measured value is 3.26µN so the estimate is not too good. For heavier nuclei the estimate
from the shell model gets much worse.
The precise origin of the magnetic dipole moment is not understood, but in general they
cannot be predicted from the shell model. For example for the nuclide 𝐹 (fluorine), the
measured value of the magnetic moment is 4.72µN whereas the value predicted form the
above model is −0.26µN. There are contributions to the magnetic moments from the nuclear
potential that is not well-understood.
Shell Model And Electric Quadrupole Moment
The shell model predicts that the electric quadrupole moment of an odd A nuclide is due to
the last proton in the nucleus. The quadrupole moment so measured corresponds to the
magnetic quantum number of the last proton mj = j. For L and S parallel (j = l + ½), this
corresponds to ml = l and ms = + . Hence the proton wave-function can be written as,
𝜓=
( )
.
𝑁 𝑃 (𝑐𝑜𝑠𝜃)𝑒
(1)
Where α is the spin wave-function and Nl is a normalisation factor so chosen that
∫[𝑁 𝑃 (𝑐𝑜𝑠𝜃)𝑒
(2)
] 𝑑Ω = 1
On substituting equation (1) in the expression for quadrupole moment, we get
𝑄 = ∫(3𝑧 − 𝑟 ) |𝜓| . 𝑑𝜏
= [∫ 𝑟 |𝑈(𝑟)| . 𝑑𝜏] × [∫ 𝑁 (3𝑐𝑜𝑠 𝜃 − 1)(𝑃 𝑐𝑜𝑠𝜃) . 𝑑Ω]
(3)
It has been shown that
(2l + 1) cos θ 𝑃 (cos θ) = 𝑃
And
∫ 𝑃 (cos 𝜃) 𝑑Ω =
𝑁 =
(
(
(
{
(cos θ)
)!
)(
(4)
)!
)
( )}
Hence after a little simplification
𝑄= −
𝑟̅ = −
𝑟̅
(5)
where j = l + ½
and
𝑟̅ = ∫ 𝑟 [𝑈(𝑟)] 𝑑𝑟
is the mean square radius of the orbit.
Similarly for l and S anti-parallel, the proton quadrupole moment can be written as
𝑄= −
𝑟̅ = −
𝑟̅
(6)
where j = l – ½
At the proton number 2, 8, 20, 50, 82 the quadrupole moment is zero or small. When a
new shell begins to form, the quadrupole moment is negative; as the number of protons in the
unfilled shell is increased, Q becomes positive and increases until it reaches a maximum
when the shell is about 2/3 filled; Q then decreases to zero at the magic proton numbers after
which it becomes negative. This behaviour fits well in the shell model predictions. But in
some cases the quadrupole moment is much larger than expected from the independent
particle model. The quadrupole moment determines the departure of a nucleus from spherical
symmetry. The large value of quadrupole moment therefore indicates that in some cases, the
nucleus is far from being spherical. These large quadrupole moments can be explained only if
we consider that the part of the nucleus consisting of the filled shells forms a core which can
be deformed by the nucleus in the unfilled shell. The agreement between theoretical and
experimental values is maximum when the nuclear core is considered to be spheroidal rather
than spherical. This modification leads to another nuclear model called Collective Model.
Shell Model And The Liquid Drop Model
A comparative study of the single particle shell model and the liquid drop model due
to Niels Bohr shows that both are in conflict with each other. In the shell model it has been
assumed that each nucleon moves in its orbit with the nucleus, independently of all other
nucleons. The orbit is determined by a potential energy function V(r). Each nucleon is
regarded as an independent particle and the interaction between nucleons is considered to be
a small perturbation on their action between a nucleon and the potential field. Thus the
interaction between the orbital nucleon and the rest is very weak. In the liquid drop model on
the other hand, the nucleons are considered to interact strongly with each other so that the
collective motions are possible. The following three examples speak in favour of such
collective motions:
(1)
(2)
(3)
The phenomenon of nuclear fission which can be easily described in terms of free
vibrations of liquid drop.
The large values of electric quadrupole moment in some nuclei cannot he accounted for
on the single particle shell model. The discrepancy between the theoretical and
experimental values can be explained only on the assumption that collective distortions
of the nuclear core involving many nucleons also contribute to the quadrupole moment.
The third example is the observed mean lives for gamma radiation of the electric
quadrupole type. The observed lives are approximately 100 times shorter than
estimated on the assumption that only one proton moves during the transition. This
suggests that changes in the deformation of the core are involved.
******************
Hard Work Never Fails