Nuclear Structure. Some Properties of Nuclei.

of Nuclei. Nuclear Binding Energy. Nuclear Models. Radioactivity. The Decay

Processes. Natural Radioactivity. Nuclear Reactions. Nuclear Magnetic Resonance and Magnetic Resonance Imaging.

number of protons and neutrons it contains, using the following

quantities:

A nuclide is a specific combination of atomic number and mass number that represents a nucleus.

The nuclei of all atoms of a particular element contain the same number of protons but often contain different numbers of neutrons. Nuclei related in this way are called isotopes. The isotopes of an element have the same Z value but different N and A values.

is defined in such a way that the mass of

one atom of the isotope 12C is exactly 12 u, where 1 u is equal to 1.660 539x10-27 kg.

atomic mass unit in terms of its rest-energy equivalent. For

one atomic mass unit,

Charge and Mass

The Size and Structure of Nuclei

Applying the conservation of energy principle to the system gives

small lengths are common in nuclear physics, an often-used convenient

length unit is the femtometer (fm), which is sometimes called the fermi and is defined as

Since the time of Rutherford’s scattering experiments, a multitude of other experiments have shown that most nuclei are approximately spherical and have an average radius given by

volume of a sphere is proportional to the cube of

its radius, the volume of a nucleus (assumed to be spherical) is directly proportional to A, the total number of nucleons. This proportionality suggests that all nuclei have nearly the same density. When nucleons combine to form a nucleus, they combine as though they were tightly packed spheres.

A nucleus can be modeled as a cluster of tightly packed spheres, where each sphere is a nucleon.

for a neutron–proton system. (b) Potential energy versus separation distance

for a proton–proton system. To display the difference in the curves on this scale, the height of the peak for the proton–proton curve has been exaggerated by a factor of 10.

Z for stable nuclei (black dots).

than the sum of the masses of its individual nucleons.

Therefore, the rest energy of the bound system (the nucleus) is less than the combined rest energy of the separated nucleons. This difference in energy is called the binding energy of the nucleus and can be interpreted as the energy that must be added to a nucleus to break it apart into its components. Therefore, to separate a nucleus into protons and neutrons, energy must be delivered to the system.

where M(H) is the atomic mass of the neutral hydrogen atom, mn is the mass of the neutron, M(AZX) represents the atomic mass of an atom of the isotope AZX, and the masses are all in atomic mass units.

strongly with one another and undergo frequent collisions as they

jiggle around within the nucleus. This jiggling motion is analogous to the thermally agitated motion of molecules in a drop of liquid.

Four major effects influence the binding energy of the nucleus in the liquid-drop model:

• The volume effect.

• The surface effect.

• The Coulomb repulsion effect.

• The symmetry effect.

the semiempirical binding-energy formula

semiempirical binding-energy formula (redbrown). For comparison to the theoretical curve,

experimental values for four sample nuclei are shown.

those calculated from the liquid-drop model is a function of

A.

The red spheres represent protons, and the gray spheres represent

neutrons.

λ, called the decay constant, is the probability of decay

per nucleus per second.

nuclei at t=0.
The decay rate R, which is the number

of decays per second, can be obtained by equation:

where R0= λN0 is the decay rate at t=0. The decay rate R of a sample is often referred to as its activity.

during which half of a given number of radioactive nuclei

decay.

A frequently used unit of activity is the curie (Ci), defined as

This value was originally selected because it is the approximate activity of 1 g of radium. The SI unit of activity is the becquerel (Bq):

and Y the daughter nucleus.
When the nucleus of one element

changes into the nucleus of another as happens in alpha decay, the process is called spontaneous decay.

the disintegration energy of the system

nucleus is initially at rest. After the decay, the radon

nucleus has kinetic energy KRn and momentum and the alpha particle has kinetic energy Ka and momentum .

mass is either zero (in which case it travels at

the speed of light) or very small; much recent persuasive experimental evidence suggests that the neutrino mass is not zero. Current experiments place the upper bound of the mass of the neutrino at approximately 7 eV/c2.
• It has a spin of 1/2, which allows the law of conservation of angular momentum to be satisfied in beta decay.
• It interacts very weakly with matter and is therefore very difficult to detect.

The neutrino has the following properties:

is electron capture:
In most cases, it is a K-shell electron

that is captured and the process is therefore referred to as K capture.

unstable nuclei found in nature, which give rise to natural

radioactivity, and (2) unstable nuclei produced in the laboratory through nuclear reactions, which exhibit artificial radioactivity.

nuclei, are called nuclear reactions.
the reaction energy Q associated with

a nuclear reaction as the difference between the initial and final rest energy resulting from the reaction:

The Q value for this reaction is 17.3 MeV. A reaction such as this one, for which Q is positive, is called exothermic. A reaction for which Q is negative is called endothermic.

occur is called the threshold energy.
If particles a and b

in a nuclear reaction are identical so that X and Y are also necessarily identical, the reaction is called a scattering event. If the kinetic energy of the system (a and X) before the event is the same as that of the system (b and Y) after the event, it is classified as elastic scattering. If the kinetic energy of the system after the event is less than that before the event, the reaction is described as inelastic scattering.

The total number of nucleons before the reaction (1+19=20) is equal to the total number after the reaction (16+4=20). Furthermore, the total charge is the same before (1=9) and after (8=2) the reaction.

orientations of the nuclear spin angular momentum vector and its

projections along the z axis for the case I =3/2.

the nuclear magneton

resonance. The radio-frequency magnetic field created by the coil surrounding

the sample and provided by the variable-frequency oscillator is perpendicular to the constant magnetic field created by the electromagnet. When the nuclei in the sample meet the resonance condition, the nuclei absorb energy from the radio-frequency field of the coil; this absorption changes the characteristics of the circuit in which the coil is included. Most modern NMR spectrometers use superconducting magnets at fixed field strengths and operate at frequencies of approximately 200 MHz.