Atomic nucleus

The atomic nucleus is the small, dense region consisting

of protons and neutrons at the center of an atom,
discovered in 1911 by E. Rutherford. An atom is composed
of a positively-charged nucleus, with a cloud of
negatively-charged electrons surrounding it, bound
together by electrostatic force. Almost all of the mass of an
atom is located in the nucleus, with a very small
contribution from the electron cloud. Protons and neutrons
are bound together to form a nucleus by the nuclear force.

 5Nuclear models
 5.1Liquid drop model
 5.2Shell models and other quantum models
 5.3Consistency between models


Introduction[edit]
History[edit]
Main article: Rutherford model
The nucleus was discovered in 1911, as a result of Ernest
Rutherford's efforts to test Thomson's "plum pudding
model" of the atom.[9] The electron had already been
discovered earlier by J.J. Thomson himself. Knowing that
atoms are electrically neutral, Thomson postulated that
there must be a positive charge as well. In his plum pudding
model, Thomson suggested that an atom consisted of
negative electrons randomly scattered within a sphere of
positive charge. Ernest Rutherford later devised an
experiment with his research partner Hans Geiger and with
help of Ernest Marsden, that involved the deflection
of alpha particles (helium nuclei) directed at a thin sheet of
metal foil. He reasoned that if Thomson's model were
correct, the positively charged alpha particles would easily
pass through the foil with very little deviation in their paths,
as the foil should act as electrically neutral if the negative
and positive charges are so intimately mixed as to make it
appear neutral. To his surprise, many of the particles were
deflected at very large angles. Because the mass of an alpha
particle is about 8000 times that of an electron, it became
apparent that a very strong force must be present if it could
deflect the massive and fast moving alpha particles. He
realized that the plum pudding model could not be
accurate and that the deflections of the alpha particles
could only be explained if the positive and negative charges
were separated from each other and that the mass of the
atom was a concentrated point of positive charge. This
justified the idea of a nuclear atom with a dense center of
positive charge and mass.
Nuclear models[edit]
Main article: nuclear structure
Although the standard model of physics is widely believed
to completely describe the composition and behavior of the
nucleus, generating predictions from theory is much more
difficult than for most other areas of particle physics. This is
due to two reasons:

 In principle, the physics within a nucleus can be

derived entirely from quantum chromodynamics (QCD).
 Even if the nuclear force is well constrained, a
significant amount of computational power is required to
accurately compute the properties of nuclei ab initio.

Historically, experiments have been compared to relatively

crude models that are necessarily imperfect. None of these
models can completely explain experimental data on
nuclear structure.[18]

The nuclear radius (R) is considered to be one of the basic

quantities that any model must predict. For stable nuclei
(not halo nuclei or other unstable distorted nuclei) the
nuclear radius is roughly proportional to the cube root of
the mass number (A) of the nucleus, and particularly in
nuclei containing many nucleons, as they arrange in more
spherical configurations:
The stable nucleus has approximately a constant density
and therefore the nuclear radius R can be approximated by
the following formula,

where A = Atomic mass number (the number of protons Z,

plus the number of neutrons N)
and r0 = 1.25 fm = 1.25 × 10− 15 m. In this equation, the
"constant" r0 varies by 0.2 fm, depending on the nucleus in
question, but this is less than 20% change from a
constant.[19]

In other words, packing protons and neutrons in the

nucleus gives approximately the same total size result as
packing hard spheres of a constant size (like marbles) into a
tight spherical or almost spherical bag (some stable nuclei
are not quite spherical, but are known to be prolate).[20]

Models of nuclear structure include :

Liquid drop model[edit]

Main article: Semi-empirical mass formula
Early models of the nucleus viewed the nucleus as a
rotating liquid drop. In this model, the trade-off of
long-range electromagnetic forces and relatively
short-range nuclear forces, together cause behavior which
resembled surface tension forces in liquid drops of different
sizes. This formula is successful at explaining many
important phenomena of nuclei, such as their changing
amounts of binding energy as their size and composition
changes (see semi-empirical mass formula), but it does not
explain the special stability which occurs when nuclei have
special "magic numbers" of protons or neutrons.

The terms in the semi-empirical mass formula, which can be

used to approximate the binding energy of many nuclei, are
considered as the sum of five types of energies (see below).
Then the picture of a nucleus as a drop of incompressible
liquid roughly accounts for the observed variation of
binding energy of the nucleus:

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 the
volume.

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.

Coulomb Energy. The electric repulsion between each pair

of protons in a nucleus contributes toward decreasing its
binding energy.

Asymmetry energy (also called Pauli Energy). An energy

associated with the Pauli exclusion principle. Were it not for
the Coulomb energy, the most stable form of nuclear
matter would have the same number of neutrons as
protons, since unequal numbers of neutrons and protons
imply filling higher energy levels for one type of particle,
while leaving lower energy levels vacant for the other type.

Pairing energy. An energy which is a correction term that

arises from the tendency of proton pairs and neutron pairs
to occur. An even number of particles is more stable than
an odd number.

Shell models and other quantum models[edit]

Main article: Nuclear shell model
A number of models for the nucleus have also been
proposed in which nucleons occupy orbitals, much like
the atomic orbitals in atomic physics theory. These wave
models imagine nucleons to be either sizeless point
particles in potential wells, or else probability waves as in
the "optical model", frictionlessly orbiting at high speed in
potential wells.

In the above models, the nucleons may occupy orbitals in

pairs, due to being fermions, which allows explanation
of even/odd Z and N effects well-known from
experiments. The exact nature and capacity of nuclear shells
differs from those of electrons in atomic orbitals, primarily
because the potential well in which the nucleons move
(especially in larger nuclei) is quite different from the central
electromagnetic potential well which binds electrons in
atoms. Some resemblance to atomic orbital models may be
seen in a small atomic nucleus like that of helium-4, in
which the two protons and two neutrons separately occupy
1s orbitals analogous to the 1s orbital for the two electrons
in the helium atom, and achieve unusual stability for the
same reason. Nuclei with 5 nucleons are all extremely
unstable and short-lived, yet, helium-3, with 3 nucleons, is
very stable even with lack of a closed 1s orbital shell.
Another nucleus with 3 nucleons, the triton hydrogen-3 is
unstable and will decay into helium-3 when isolated. Weak
nuclear stability with 2 nucleons {NP} in the 1s orbital is
found in the deuteron hydrogen-2, with only one nucleon
in each of the proton and neutron potential wells. While
each nucleon is a fermion, the {NP} deuteron is a boson and
thus does not follow Pauli Exclusion for close packing within
shells. Lithium-6 with 6 nucleons is highly stable without a
closed second 1p shell orbital. For light nuclei with total
nucleon numbers 1 to 6 only those with 5 do not show
some evidence of stability. Observations of beta-stability of
light nuclei outside closed shells indicate that nuclear
stability is much more complex than simple closure of shell
orbitals with magic numbers of protons and neutrons.

For larger nuclei, the shells occupied by nucleons begin to

differ significantly from electron shells, but nevertheless,
present nuclear theory does predict the magic numbers of
filled nuclear shells for both protons and neutrons. The
closure of the stable shells predicts unusually stable
configurations, analogous to the noble group of
nearly-inert gases in chemistry. An example is the stability
of the closed shell of 50 protons, which allows tin to have 10
stable isotopes, more than any other element. Similarly, the
distance from shell-closure explains the unusual instability
of isotopes which have far from stable numbers of these
particles, such as the radioactive elements 43 (technetium)
and 61 (promethium), each of which is preceded and
followed by 17 or more stable elements.

There are however problems with the shell model when an

attempt is made to account for nuclear properties well away
from closed shells. This has led to complex post
hoc distortions of the shape of the potential well to fit
experimental data, but the question remains whether these
mathematical manipulations actually correspond to the
spatial deformations in real nuclei. Problems with the shell
model have led some to propose realistic two-body and
three-body nuclear force effects involving nucleon clusters
and then build the nucleus on this basis. Three such cluster
models are the 1936 Resonating Group Structure model of
John Wheeler, Close-Packed Spheron Model of Linus
Pauling and the 2D Ising Model of MacGregor.