Binding energy is the
mechanical energy required to disassemble
a whole into separate parts. A bound system has typically a lower
potential energy than its
constituent parts; this is what keeps the system together. The
usual convention is that this corresponds to a
positive
binding energy.
In general, binding energy represents the
mechanical work which must be done in acting
against the forces which hold an object together, while
disassembling the object into component parts separated by
sufficient distance that further separation requires negligible
additional work.
At the
atomic level the
atomic binding energy of the atom derives from
electromagnetic
interaction and is the
energy
required to disassemble an atom into free electrons and a
nucleus.
Electron binding energy is a
measure of the energy required to free electrons from their atomic
orbits.
At the
nuclear level the
nuclear binding energy (binding energy of
nucleons into a
nuclide) is
derived from the
strong nuclear
force and is the
energy required to
disassemble a
nucleus into the same
number of free unbound
neutrons and
protons it is composed of, in such a way that the
particles are far/distant enough from each other so that the strong
nuclear force can no longer cause the particles to interact.
In
astrophysics,
gravitational binding energy of
a celestial body is the energy required to expand the material to
infinity. This quantity is not to be confused with the
gravitational potential
energy, which is the energy required to separate two bodies,
such as a celestial body and a satellite, to infinite distance,
keeping each intact (the latter energy is lower).
In bound systems, if the binding energy is removed from the system,
it must be subtracted from the mass of the unbound system, simply
because this energy has mass, and if subtracted from the system at
the time it is bound, will result in removal of mass from the
system. System mass is not conserved in this process because the
system is not
closed during the binding
process.
Mass deficit
Classically a bound system is at a lower energy level than its
unbound constituents, its mass must be less than the total mass of
its unbound constituents. For systems with low binding energies,
this "lost" mass after binding may be fractionally small. For
systems with high binding energies, however, the missing mass may
be an easily measurable fraction.
Since all forms of energy have mass, the question of where the
missing mass of the binding energy goes, is of interest. The answer
is that this mass is lost from a system which is not closed. It
transforms to heat, light, higher energy states of the nucleus/atom
or other forms of energy, but these types of energy also have mass,
and it is necessary that they be removed from the system before its
mass may decrease. The "mass deficit" from binding energy is
therefore removed mass that corresponds with removed energy,
according to Einstein's equation E = mc
^{2}. Once
the system cools to normal temperatures and returns to ground
states in terms of energy levels, there is less mass remaining in
the system than there was when it first combined and was at high
energy. Mass measurements are almost always made at low
temperatures with systems in ground states, and this difference
between the mass of a system and the sum of the masses of its
isolated parts is called a mass deficit. Thus, if binding energy
mass is transformed into heat, the system must be cooled (the heat
removed) before the mass-deficit appears in the cooled system. In
that case, the removed heat represents exactly the mass "deficit",
and the heat itself retains the mass which was lost.
As an illustration, consider two objects attracting each other in
space through their
gravitational
field. The attraction force accelerates the objects and they
gain some speed toward each other converting the potential
(gravity) energy into kinetic (movement) energy. When either the
particles 1) pass through each other without interaction or 2)
elastically repel during the collision, the gained kinetic energy
(related to speed), starts to revert into potential form driving
the collided particles apart. The decelerating particles will
return to the initial distance and beyond into infinity or stop and
repeat the collision (oscillation takes place). This shows that the
system, which loses no energy, does not combine (bind) into a solid
object, parts of which oscillate at short distances. Therefore, in
order to bind the particles, the kinetic energy gained due to the
attraction must be dissipated (by resistive force). Complex objects
in collision ordinarily undergo
inelastic collision, transforming some
kinetic energy into internal energy (heat content, which is atomic
movement), which is further radiated in the form of photons—the
light and heat. Once the energy to escape the gravity is dissipated
in the collision, the parts will oscillate at closer, possibly
atomic, distance, thus looking like one solid object. This lost
energy, necessary to overcome the potential barrier in order to
separate the objects, is the binding energy. If this binding energy
were retained in the system as heat, its mass would not decrease.
However, binding energy lost from the system (as heat radiation)
would itself have mass, and directly represent of the "mass
deficit" of the cold, bound system.
Closely analogous considerations apply in chemical and nuclear
considerations. Exothermic chemical reactions in closed systems do
not change mass, but (in theory) become less massive once the heat
of reaction is removed. This mass change is too small to measure
with standard equipment. In nuclear reactions, however, the
fraction of mass that may be removed as light or heat, i.e.,
binding energy, is often a much larger fraction of the system mass.
It may thus be measured directly as a mass difference between rest
masses of reactants and products. This is because nuclear forces
are comparatively stronger than Coulombic forces associated with
the interactions between electrons and protons that generate heat
in chemistry.
Mass defect
In simple words definition of mass defect can be stated as
follows:
Definition: The distance between theoretical calculated mass and
experimentally measured mass of nucleus is called mass defect. It
is denoted by Δm. It can be calculated as follows:
- Mass defect = (Theoretical calculated mass) - (measured mass of
nucleus)
- : i.e, (sum of masses of protons and neutrons) - (measured mass
of nucleus)
In nuclear reactions, the energy that must be
radiated or otherwise removed as binding energy
may be in the form of electromagnetic waves, such as
gamma radiation, or as heat. Again, however,
no mass deficit can in theory appear until this radiation has been
emitted and is no longer part of the system.
The energy given off during either
nuclear fusion or
nuclear fission is the difference between
the binding energies of the fuel and the fusion or fission
products. In practice, this energy may also be calculated from the
substantial mass differences between the fuel and products, once
evolved heat and radiation have been removed.
When the nucleons are grouped together to form a nucleus, they lose
a small amount of mass, i.e., there is mass defect. This mass
defect is released as (often radiant) energy according to the
relation E = mc
^{2}; thus
binding energy =
mass defect · c^{2} .This energy holds the
nucleons together and is known as binding energy. In fact, mass
defect is a measure of the binding energy of the nucleus. The
greater the mass defect, the greater is the binding energy of the
nucleus.
Mass excess
It is observed experimentally that the mass of the nucleus is
smaller than the number of nucleons each counted with a mass of 1
a.m.u.. This difference is called
mass excess.
The difference between the actual mass of the nucleus measured
in atomic mass units and the
number of nucleons is called mass excess i.e.
Mass excess = M - A = Excess-energy /
c^2
with M equals the actual mass of the nucleus in
uand A equals the
mass number.
This mass excess is a practical value calculated from
experimentally measured nucleon masses and stored in nuclear
databases. For middle-weight nuclides this value is negative in
contrast to the mass defect which is never negative for any
nuclide.
Nuclear binding energy
Practice: Binding energy for atoms
Definition: The amount of energy required to break the nucleus of
an atom into its isolated nucleons is called Nuclear binding
energy.The measured mass deficits of
isotopes are always listed as mass deficits of the
neutral atoms
of that isotope, and mostly in
MeV. As a
consequence, the listed mass deficits are not a measure for the
stability or binding energy of isolated nuclei, but for the whole
atoms. This has very practical reasons, because it is very hard to
totally
ionize heavy elements, i.e. strip them
of all of their
electrons.
This
practice is useful for other reasons, too: Stripping all the
electrons from a heavy unstable nucleus (thus producing a bare
nucleus) will change the lifetime of the nucleus, indicating that
the nucleus cannot be treated independently (Experiments at the
heavy ion accelerator GSI). This is also evident from
phenomena like
electron capture. Theoretically, in orbital
models of heavy atoms, the electron orbits partially inside the
nucleus (it doesn't
orbit in a strict
sense, but has a non-vanishing probability of being located inside
the nucleus).
Of course, a
nuclear decay happens to
the nucleus, meaning that properties ascribed to the nucleus will
change in the event. But for the following considerations and
examples, you should keep in mind that "mass deficit" as a measure
for "binding energy", and as listed in nuclear data tables, means
"mass deficit of the neutral atom" and is a measure for stability
of the whole atom.
Nuclear binding energy curve
300 px
In the
periodic table of
elements, the series of light elements from
hydrogen up to
sodium is
observed to exhibit generally increasing binding energy per nucleon
as the
atomic mass increases. This
increase is generated by increasing forces per nucleon in the
nucleus, as each additional nucleon is attracted by all of the
other nucleons, and thus more tightly bound to the whole.
The region of increasing binding energy is followed by a region of
relative stability (saturation) in the sequence from
magnesium through
xenon. In
this region, the nucleus has become large enough that nuclear
forces no longer completely extend efficiently across its width.
Attractive nuclear forces in this region, as atomic mass increases,
are nearly balanced by repellent electromagnetic forces between
protons, as
atomic number
increases.
Finally, in elements heavier than xenon, there is a decrease in
binding energy per nucleon as atomic number increases. In this
region of nuclear size, electromagnetic repulsive forces are
beginning to gain against the strong nuclear force.
At the peak of binding energy,
nickel-62
is the most tightly-bound nucleus (per nucleon), followed by
iron-58 and
iron-56.
This is the approximate basic reason why iron and nickel are very
common metals in planetary cores, since they are produced profusely
as end products in
supernovae and in the
final stages of
silicon burning in
stars. However, it is not binding energy per defined nucleon (as
defined above) which controls which exact nuclei are made, because
within stars, neutrons are free to convert to protons to release
even more energy, per generic nucleon, if the result is a stable
nucleus with a larger fraction of protons. Thus, iron-56 has most
binding energy of any group of 56 nucleons (because of its
relatively larger fraction of protons), even while having less
binding energy per nucleon than nickel-62, if this binding energy
is computed by comparing Ni-62 with its disassembly products of 28
protons and 34 neutrons. In fact, it has been argued that
photodisintegration of
^{62}Ni
to form
^{56}Fe may be energetically possible in an
extremely hot star core, due to this beta decay conversion of
neutrons to protons.
It is generally believed that iron-56 is more common than nickel
isotopes in the universe for mechanistic reasons, because its
unstable progenitor
nickel-56 is copiously
made by staged build-up of 14 helium nuclei inside supernovas,
where it has no time to decay to iron before being released into
the interstellar medium in a matter of a few minutes as a star
explodes. However, nickel-56 then decays to
iron-56 within a few weeks. The gamma ray light
curve of such a process has been observed to happen in type IIa
supernovae, such as
SN1987a. In a star,
there are no good ways to create nickel-62 by alpha-addition
processes, or there might be more of it in the universe.
Measuring the binding energy
The existence of a
maximum in binding energy in
medium-sized nuclei is a consequence of the trade-off in the
effects of two opposing forces which have different range
characteristics. The attractive nuclear force (
strong nuclear force), which binds
protons and neutrons equally to each other, has a limited range due
to a rapid exponential decrease in this force with distance.
However, the repelling electromagnetic force, which acts between
protons to force nuclei apart, falls off with distance much more
slowly (as the inverse square of distance). For nuclei larger than
about four nucleons in diameter, the additional repelling force of
additional protons more than offsets any binding energy which
results between further added nucleons as a result of additional
strong force interactions; such nuclei become less and less tightly
bound as their size increases, though most of them are still
stable. Finally, nuclei containing more than 209 nucleons (larger
than about 6 nucleons in diameter) are all too large to be stable,
and are subject to spontaneous decay to smaller nuclei.
Nuclear fusion produces energy by
combining the very lightest elements into more tightly-bound
elements (such as hydrogen into
helium), and
nuclear fission produces energy by
splitting the heaviest elements (such as
uranium and
plutonium) into
more tightly-bound elements (such as
barium
and
krypton). Both processes produce energy,
because middle-sized nuclei are the most tightly bound of
all.
As seen above in the example of deuterium, nuclear binding energies
are large enough that they may be easily measured as fractional
mass deficits, according to the equivalence of
mass and energy. The atomic binding energy is simply the amount of
energy (and mass) released, when a collection of free
nucleons are joined together to form a
nucleus.
Nuclear binding energy can be easily computed from the easily
measurable difference in mass of a nucleus, and the sum of the
masses of the number of free neutrons and protons that make up the
nucleus. Once this mass difference, called the
mass
defect or
mass deficiency, is known,
Einstein's
mass-energy
equivalence formula
E =
mc² can be
used to compute the binding energy of any nucleus. (As a historical
note, early nuclear physicists used to refer to computing this
value as a "packing fraction" calculation.)
For example, the
atomic mass unit
(1
u) is
defined to be 1/12 of the mass
of a
^{12}C atom—but the atomic mass of a
^{1}H
atom (which is a proton plus electron) is 1.007825
u, so each nucleon in
^{12}C has lost, on
average, about 0.8% of its mass in the form of binding
energy.
Semiempirical formula for nuclear binding energy
For a nucleus with A nucleons including Z protons, a semiemipirical
formula for the binding energy (
B.E.) per nucleon
(
A)
is:B.E./A=a-b/A^{1/3}-cZ^2/A^{4/3}-d(N-Z)^2/A^2\pm e/A^{7/4}
where the binding energy is in MeV for the following numerical
values of the constants: a=14.0; b=13.0; c=0.585; d=19.3;
e=33.
The first term a\, is called the saturation contribution and
ensures that the B.E. per nucleon is the same for all nuclei to a
first approximation. The term -b/A^{1/3}\, is a surface tension
effect and is proportional to the number of nucleons that are
situated on the nuclear surface; it is largest for light nuclei.
The term -cZ^2/A^{4/3}\, is the Coulomb electrostatic repulsion;
this becomes more important as Z increases. The symmetry correction
term -d(N-Z)^2/A^2\, takes into account the fact that in the
absence of other effects the most stable arrangement has equal
numbers of protons and neutrons; this is because the
n-p
interaction in a nucleus is stronger than either the
n-n
or
p-p interaction. The pairing term \pm e/A^{7/4} is
purely empirical; it is + for even-even nuclei and - for
odd-odd nuclei.
Example values deduced from experimentally measured atom
nuclide masses
All
mass excess data are taken from .
Notice also that we use 1 u = 1
a.m.u = 931.494028(±0.000023) MeV. To
calculate the "binding energy" we use the formula
P*(m
_{p}+m
_{e}) + N * m
_{n} -
m
_{nuclide} where P denotes the number of protons of the
nuclides and N its number of neutrons. We takem
_{p} =
938.2723 Mev, m
_{e} = 0.5110 MeV and m
_{n} =
939.5656 MeV. The letter A denotes the sum of P and N (number of
nucleons in the nuclide). If we assume the reference nucleon has
the mass of a neutron (so that all "total" binding energies
calculated are maximal) we could define the total binding energy as
the difference from the mass of the nucleus, and the mass of a
collection of A free neutrons. In other words, it would be [(P+N)*
m
_{n}] - m
_{nuclide}. The "
total
binding energy per nucleon" would be this value divided by A.
most strongly bound nuclides atoms
nuclide |
P |
N |
mass excess |
total mass |
total mass / A |
total binding energy / A |
mass defect |
binding energy |
binding energy / A |
^{56}Fe |
26 |
30 |
-60.6054 MeV |
55.934937 u |
0.9988382 u |
9.1538 MeV |
0.528479 u |
492.275 MeV |
8.7906 MeV |
^{58}Fe |
26 |
32 |
-62.1534 MeV |
57.933276 u |
0.9988496 u |
9.1432 MeV |
0.547471 u |
509.966 MeV |
8.7925 MeV |
^{60}Ni |
28 |
32 |
-64.4721 MeV |
59.930786 u |
0.9988464 u |
9.1462 MeV |
0.565612 u |
526.864 MeV |
8.7811 MeV |
^{62}Ni |
28 |
34 |
-66.7461 MeV |
61.928345 u |
0.9988443 u |
9.1481 MeV |
0.585383 u |
545.281 MeV |
8.7948 MeV |
In this calculation
^{56}Fe has the lowest nucleon-specific
mass of the four nuclides, but this does not mean it is the
strongest bound atom per hadron, unless the choice of beginning
hadrons is completely free. Iron releases the largest energy if any
56 nucleons are allowed to build a nuclide—changing one to another
if necessary, The highest "binding energy" per hadron, with the
hadrons starting as the same number of protons Z and total nucleons
A as in the bound nucleus, is
^{62}Ni. Thus, the true
absolute value of the total binding energy of a nucleus depends on
what we are "allowed" to construct the nucleus out of. If all
nuclei of mass number A were to be allowed to be constructed of A
neutrons, then Fe-56 would release the most energy per nucleon,
since it has a larger fraction of protons than Ni-62. However, if
nucleons are required to be constructed of only the same number of
protons and neutrons that they contain, then nickel-62 is the most
tightly bound nucleus, per nucleon.
some light nuclides resp. atoms
nuclide |
P |
N |
mass excess |
total mass |
total mass / A |
total binding energy / A |
mass defect |
binding energy |
binding energy / A |
n |
0 |
1 |
8.0716 MeV |
1.008665 u |
1.008665 u |
0.0000 MeV |
0 u |
0 MeV |
0 MeV |
^{1}H |
1 |
0 |
7.2890 MeV |
1.007825 u |
1.007825 u |
0.7826 MeV |
0.0000000146 u |
0.0000136 MeV |
13.6 eV |
^{2}H |
1 |
1 |
13.13572 MeV |
2.014102 u |
1.007051 u |
1.50346 MeV |
0.002388 u |
2.22452 MeV |
1.11226 MeV |
^{3}H |
1 |
2 |
14.9498 MeV |
3.016049 u |
1.005350 u |
3.08815 MeV |
0.0091058 u |
8.4820 MeV |
2.8273 MeV |
^{3}He |
2 |
1 |
14.9312 MeV |
3.016029 u |
1.005343 u |
3.09433 MeV |
0.0082857 u |
7.7181 MeV |
2.5727 MeV |
In the table above it can be seen that the decay of a neutron, as
well as the transformation of tritium into helium-3, releases
energy; hence, it manifests a stronger bound new state when
measured against the mass of an equal number of neutrons (and also
a lighter state per number of total hadrons). Such reactions are
not driven by changes in binding energies as calculated from
perviously fixed N and Z numbers of neutrons and protons, but
rather in decreases in the total mass of the nuclide/per nucleon,
with the reaction.
References
External links
See also