Mass excess
Encyclopedia
The mass excess of a nuclide
is the difference between its actual mass and its mass number
in atomic mass units. It is one of the predominant methods for tabulating nuclear mass. The mass of an atomic nucleus
is well approximated by its mass number, which indicates that most of the mass of a nucleus arises from mass of its constituent proton
s and neutron
s. Thus, the mass excess is an expression of the nuclear binding energy
, relative to the binding energy per nucleon
of Carbon-12
(which defines the atomic mass unit). If the mass excess is negative, the nucleus has more binding energy than 12C, and vice versa. If a nucleus has a large excess of mass compared to a nearby nuclear species, it can radioactively decay
, releasing energy.
or decay. From Einstein's
work on mass-energy equivalence
(E=mc2), it is well known that nuclear reaction
s transform energy into mass and vice versa. However, most of the energy remains in the mass of the nuclei involved, and only a small fraction of the total energy, on the order of 0.01 to 0.1% of the total mass, may be absorbed or liberated. Thus by working in terms of the mass excess, one has effectively removed much of the mass changes which arise from the mere transfer or release of nucleons, making more obvious the scale of the net energy difference.
Nuclear reaction kinematics are customarily performed in units involving the electron volt
, a consequence of accelerator
technology. The combination of this practical point with the theoretical relation E=mc2 makes units of mega electron volts over the speed of light squared (MeV/c2) a convenient form to express nuclear mass. However, the numerical values of nuclear masses in MeV/c2 are quite large (even the proton mass is ~938.27 MeV/c2), while mass excesses range in the tens of MeV/c2. This makes tabulated mass excess less cumbersome for use in calculations. One trivial point worth noting that the 1/c2 term is typically omitted when quoting mass excess values in MeV, since the interest is more often energy and not mass; if one wanted units of mass, one would simply change the units from MeV to MeV/c2 without altering the numerical value.
of 236U into 92Kr, 141Ba, and three neutrons.
236U → 92Kr + 141Ba + 3 n
The mass number of 236U is 236 u (atomic mass units), but the actual mass is
u, so the mass excess is 
u. Calculated in the same manner, the mass excess for 92Kr, 141Ba, and a neutron are 
u, 
u and 
u, respectively. The mass excess of the reactant is 
u, and the mass excess of the products is 
The difference between reactants and products is
u, which shows that the mass excess of the products is less than that of the reactants, and so the fission can occur.
The resulting difference in mass excess can be converted into energy using 1 u = 931.494 MeV/c², yielding 166 MeV.
Nuclide
A nuclide is an atomic species characterized by the specific constitution of its nucleus, i.e., by its number of protons Z, its number of neutrons N, and its nuclear energy state....
is the difference between its actual mass and its mass number
Mass number
The mass number , also called atomic mass number or nucleon number, is the total number of protons and neutrons in an atomic nucleus. Because protons and neutrons both are baryons, the mass number A is identical with the baryon number B as of the nucleus as of the whole atom or ion...
in atomic mass units. It is one of the predominant methods for tabulating nuclear mass. The mass of an atomic nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
is well approximated by its mass number, which indicates that most of the mass of a nucleus arises from mass of its constituent proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s and neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s. Thus, the mass excess is an expression of the nuclear binding energy
Nuclear binding energy
Nuclear binding energy is the energy required to split a nucleus of an atom into its component parts. The component parts are neutrons and protons, which are collectively called nucleons...
, relative to the binding energy per nucleon
Nucleon
In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...
of Carbon-12
Carbon-12
Carbon-12 is the more abundant of the two stable isotopes of the element carbon, accounting for 98.89% of carbon; it contains 6 protons, 6 neutrons, and 6 electrons....
(which defines the atomic mass unit). If the mass excess is negative, the nucleus has more binding energy than 12C, and vice versa. If a nucleus has a large excess of mass compared to a nearby nuclear species, it can radioactively decay
Radioactive decay
Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...
, releasing energy.
Energy Scale of Nuclear Reactions
The 12C standard makes it useful to think about nuclear mass in atomic mass units for the definition of the mass excess. However, its usefulness arises in the calculation of nuclear reaction kinematicsKinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....
or decay. From Einstein's
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
work on mass-energy equivalence
Mass-energy equivalence
In physics, mass–energy equivalence is the concept that the mass of a body is a measure of its energy content. In this concept, mass is a property of all energy, and energy is a property of all mass, and the two properties are connected by a constant...
(E=mc2), it is well known that nuclear reaction
Nuclear reaction
In nuclear physics and nuclear chemistry, a nuclear reaction is semantically considered to be the process in which two nuclei, or else a nucleus of an atom and a subatomic particle from outside the atom, collide to produce products different from the initial particles...
s transform energy into mass and vice versa. However, most of the energy remains in the mass of the nuclei involved, and only a small fraction of the total energy, on the order of 0.01 to 0.1% of the total mass, may be absorbed or liberated. Thus by working in terms of the mass excess, one has effectively removed much of the mass changes which arise from the mere transfer or release of nucleons, making more obvious the scale of the net energy difference.
Nuclear reaction kinematics are customarily performed in units involving the electron volt
Electronvolt
In physics, the electron volt is a unit of energy equal to approximately joule . By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an electric potential difference of one volt...
, a consequence of accelerator
Particle accelerator
A particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field accelerators.In...
technology. The combination of this practical point with the theoretical relation E=mc2 makes units of mega electron volts over the speed of light squared (MeV/c2) a convenient form to express nuclear mass. However, the numerical values of nuclear masses in MeV/c2 are quite large (even the proton mass is ~938.27 MeV/c2), while mass excesses range in the tens of MeV/c2. This makes tabulated mass excess less cumbersome for use in calculations. One trivial point worth noting that the 1/c2 term is typically omitted when quoting mass excess values in MeV, since the interest is more often energy and not mass; if one wanted units of mass, one would simply change the units from MeV to MeV/c2 without altering the numerical value.
Example
Consider the nuclear fissionNuclear fission
In nuclear physics and nuclear chemistry, nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts , often producing free neutrons and photons , and releasing a tremendous amount of energy...
of 236U into 92Kr, 141Ba, and three neutrons.
236U → 92Kr + 141Ba + 3 n
The mass number of 236U is 236 u (atomic mass units), but the actual mass is
u, so the mass excess is u. Calculated in the same manner, the mass excess for 92Kr, 141Ba, and a neutron are u, u and u, respectively. The mass excess of the reactant is u, and the mass excess of the products is The difference between reactants and products is
u, which shows that the mass excess of the products is less than that of the reactants, and so the fission can occur.The resulting difference in mass excess can be converted into energy using 1 u = 931.494 MeV/c², yielding 166 MeV.