The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an
atom, discovered in 1911 by Ernest Rutherford at the University of Manchester based on the 1909
Geiger–Marsden gold foil experiment. After the discovery of the neutron in 1932, models for a
nucleus composed of protons and neutrons were quickly developed by Dmitri Ivanenko[1] and
Werner Heisenberg.[2][3][4][5][6] 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.
The diameter of the nucleus is in the range of 1.70 fm (1.70×10−15 m[7]) for hydrogen (the diameter
of a single proton) to about 11.7 fm for uranium.[8] These dimensions are much smaller than the
diameter of the atom itself (nucleus + electron cloud), by a factor of about 26,634 (uranium atomic
radius is about 156 pm (156×10−12 m))[9] to about 60,250 (hydrogen atomic radius is about 52.92
pm).[a]
The branch of physics involved with the study and understanding of the atomic nucleus, including its
composition and the forces that bind it together, is called nuclear physics.
History
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.[10] The electron had already been discovered by J. J. Thomson.
Knowing that atoms are electrically neutral, J. J. 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 J. J. 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.
Etymology
The term nucleus is from the Latin word nucleus, a diminutive of nux ('nut'), meaning 'the kernel'
(i.e., the 'small nut') inside a watery type of fruit (like a peach). In 1844, Michael Faraday used the
term to refer to the "central point of an atom". The modern atomic meaning was proposed by Ernest
Rutherford in 1912.[11] The adoption of the term "nucleus" to atomic theory, however, was not
immediate. In 1916, for example, Gilbert N. Lewis stated, in his famous article The Atom and the
Molecule, that "the atom is composed of the kernel and an outer atom or shell."[12] Similarly, the
term kern meaning kernel is used for nucleus in German and Dutch.
Principles
A figurative depiction of the helium-4 atom with the electron cloud in shades of gray. In the nucleus,
the two protons and two neutrons are depicted in red and blue. This depiction shows the particles as
separate, whereas in an actual helium atom, the protons are superimposed in space and most likely
found at the very center of the nucleus, and the same is true of the two neutrons. Thus, all four
particles are most likely found in exactly the same space, at the central point. Classical images of
separate particles fail to model known charge distributions in very small nuclei. A more accurate
image is that the spatial distribution of nucleons in a helium nucleus is much closer to the helium
electron cloud shown here, although on a far smaller scale, than to the fanciful nucleus image. Both
the helium atom and its nucleus are spherically symmetric.
The nucleus of an atom consists of neutrons and protons, which in turn are the manifestation of
more elementary particles, called quarks, that are held in association by the nuclear strong force in
certain stable combinations of hadrons, called baryons. The nuclear strong force extends far enough
from each baryon so as to bind the neutrons and protons together against the repulsive electrical
force between the positively charged protons. The nuclear strong force has a very short range, and
essentially drops to zero just beyond the edge of the nucleus. The collective action of the positively
charged nucleus is to hold the electrically negative charged electrons in their orbits about the
nucleus. The collection of negatively charged electrons orbiting the nucleus display an affinity for
certain configurations and numbers of electrons that make their orbits stable. Which chemical
element an atom represents is determined by the number of protons in the nucleus; the neutral
atom will have an equal number of electrons orbiting that nucleus. Individual chemical elements can
create more stable electron configurations by combining to share their electrons. It is that sharing of
electrons to create stable electronic orbits about the nuclei that appears to us as the chemistry of
our macro world.
Protons define the entire charge of a nucleus, and hence its chemical identity. Neutrons are
electrically neutral, but contribute to the mass of a nucleus to nearly the same extent as the protons.
Neutrons can explain the phenomenon of isotopes (same atomic number with different atomic
mass). The main role of neutrons is to reduce electrostatic repulsion inside the nucleus.
Composition and shape
Protons and neutrons are fermions, with different values of the strong isospin quantum number, so
two protons and two neutrons can share the same space wave function since they are not identical
quantum entities. They are sometimes viewed as two different quantum states of the same particle,
the nucleon.[13][14] Two fermions, such as two protons, or two neutrons, or a proton + neutron
(the deuteron) can exhibit bosonic behavior when they become loosely bound in pairs, which have
integer spin.
In the rare case of a hypernucleus, a third baryon called a hyperon, containing one or more strange
quarks and/or other unusual quark(s), can also share the wave function. However, this type of
nucleus is extremely unstable and not found on Earth except in high-energy physics experiments.
The neutron has a positively charged core of radius ≈ 0.3 fm surrounded by a compensating negative
charge of radius between 0.3 fm and 2 fm. The proton has an approximately exponentially decaying
positive charge distribution with a mean square radius of about 0.8 fm.[15]
The shape of the atomic nucleus can be spherical, rugby ball-shaped (prolate deformation), discus-
shaped (oblate deformation), triaxial (a combination of oblate and prolate deformation) or pear-
shaped.[16][17]
Forces
Nuclei are bound together by the residual strong force (nuclear force). The residual strong force is a
minor residuum of the strong interaction which binds quarks together to form protons and
neutrons. This force is much weaker between neutrons and protons because it is mostly neutralized
within them, in the same way that electromagnetic forces between neutral atoms (such as van der
Waals forces that act between two inert gas atoms) are much weaker than the electromagnetic
forces that hold the parts of the atoms together internally (for example, the forces that hold the
electrons in an inert gas atom bound to its nucleus).
The nuclear force is highly attractive at the distance of typical nucleon separation, and this
overwhelms the repulsion between protons due to the electromagnetic force, thus allowing nuclei
to exist. However, the residual strong force has a limited range because it decays quickly with
distance (see Yukawa potential); thus only nuclei smaller than a certain size can be completely
stable. The largest known completely stable nucleus (i.e. stable to alpha, beta, and gamma decay) is
lead-208 which contains a total of 208 nucleons (126 neutrons and 82 protons). Nuclei larger than
this maximum are unstable and tend to be increasingly short-lived with larger numbers of nucleons.
However, bismuth-209 is also stable to beta decay and has the longest half-life to alpha decay of any
known isotope, estimated at a billion times longer than the age of the universe.
The residual strong force is effective over a very short range (usually only a few femtometres (fm);
roughly one or two nucleon diameters) and causes an attraction between any pair of nucleons. For
example, between a proton and a neutron to form a deuteron [NP], and also between protons and
protons, and neutrons and neutrons.
Halo nuclei and nuclear force range limits
The effective absolute limit of the range of the nuclear force (also known as residual strong force) is
represented by halo nuclei such as lithium-11 or boron-14, in which dineutrons, or other collections
of neutrons, orbit at distances of about 10 fm (roughly similar to the 8 fm radius of the nucleus of
uranium-238). These nuclei are not maximally dense. Halo nuclei form at the extreme edges of the
chart of the nuclides—the neutron drip line and proton drip line—and are all unstable with short
half-lives, measured in milliseconds; for example, lithium-11 has a half-life of 8.8 ms.
Halos in effect represent an excited state with nucleons in an outer quantum shell which has unfilled
energy levels "below" it (both in terms of radius and energy). The halo may be made of either
neutrons [NN, NNN] or protons [PP, PPP]. Nuclei which have a single neutron halo include 11Be and
19C. A two-neutron halo is exhibited by 6He, 11Li, 17B, 19B and 22C. Two-neutron halo nuclei break
into three fragments, never two, and are called Borromean nuclei because of this behavior (referring
to a system of three interlocked rings in which breaking any ring frees both of the others). 8He and
14Be both exhibit a four-neutron halo. Nuclei which have a proton halo include 8B and 26P. A two-
proton halo is exhibited by 17Ne and 27S. Proton halos are expected to be more rare and unstable
than the neutron examples, because of the repulsive electromagnetic forces of the halo proton(s).
Nuclear models
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). In practice however, current computational and mathematical approaches for solving QCD in
low-energy systems such as the nuclei are extremely limited. This is due to the phase transition that
occurs between high-energy quark matter and low-energy hadronic matter, which renders
perturbative techniques unusable, making it difficult to construct an accurate QCD-derived model of
the forces between nucleons. Current approaches are limited to either phenomenological models
such as the Argonne v18 potential or chiral effective field theory.[18]
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. Developments in many-body
theory have made this possible for many low mass and relatively stable nuclei, but further
improvements in both computational power and mathematical approaches are required before
heavy nuclei or highly unstable nuclei can be tackled.
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.[19]
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,
{\displaystyle R=r_{0}A^{1/3}\,}
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.[20]
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).[21]
Models of nuclear structure include:
Cluster model
The cluster model describes the nucleus as a molecule-like collection of proton-neutron groups (e.g.,
alpha particles) with one or more valence neutrons occupying molecular orbitals.[22][23][24][25]
Liquid drop model
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
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.
