Nuclear Mass and Binding Energy (English)

2-3: Mass of Nuclei, Binding Energy

A nucleus consists of

Z

protons and

N

neutrons.

Let the mass of the nucleus be

M (Z, N ).

If a nucleus is assumed

to be a simple

collection of Z

protons and N

neutrons,

the mass of the nucleus

would be just the sum of

the masses of

these constituent nucleons,

i.e. ,

where is the mass of proton and is that of neutron.

However, a nucleus

is not a simple collection of

protons and neutrons (nucleons),

but they strongly combine

with each other

through a strong interaction

named the nuclear force.

[Mass Defect of Nuclei]

In general,

if two or more particles interact

to combin together,

then the total mass of the system

would decrease to be less than

the sum of the masses of

the individual particles.

The stronger the interaction becomes,

the more the mass decreases.

This decrease of the mass of the system

is called the mass defect.

The mass defect of a nucleus

of proton number Z

and neutron number N

is defined by

[Mass-Energy Equivalence, Unit of Mass]

According to

the Special Theory of Relativity

proposed by

A. Einstein

(Germany, USA: 1879-1955),

it was made clear that

mass and energy

are equivalent

(the Mass-Energy Equivalence),

which is expressed by

an equality

On the right-hand side of this

equality,

mass is multiplied by the square

of the speed of light.

This implies that

very small amounts of mass

may be converted into

a very large amount of

energy, and vice versa.

In the World of the Atomic Nucleus,

a usually treated mass is, in general,

very small. Therefore mass in this world

is sometimes expressed

in units of energy

by being converted into

energy with the above-mentioned

Einstein's mass-energy equivalence.

For example,

the proton mass is

which is a quite small amount.

Hence, this is converted

into energy by being multiplied

by the square of c

i.e.

Accordingly,

the proton mass is expressed as

In this way,

the unit of mass

is often used

in the World of the Atomic Nucleus.

[Nuclear Binding Energies]

Since a nucleus consists

of A

nucleons

(Z

protons and N

neutrons;

A = Z + N

)

being bound together,

the mass defect

occurs

as explained at the top

of the present page.

The binding energy

B (Z, N )

is what this mass defect

is converted into

with the mass-energy equivalence.

It is therefore

given by

where

M (Z, N )

is the mass of a nuclide

of Z

protons and N

neutrons.

Namely,

the binding energy denotes

the degree

how strongly the nucleons

bind together,

and its value can be obtained

by measuring

the mass of the nuclide.

Nowadays, we can obtain

the precise value of

the mass of an individual nuclide

using the mass spectrometer

for example.

Experiments to measure

the nuclear binding energies

of wide range of nuclides

have been carried out.

The resultant data

are shown in the figure below.

From this figure, we can see that,

the binding energy per nucleon

is nearly constant

( = about 8 MeV)

for the nuclei of A

> 20,

though there are some irregularities

in the region of light nuclei.

The property that

the binding energy per nucleon

is nearly constant

is called

the saturation of the nuclear binding energy.

This is one of the remarkable

properties of the atomic nuclei.

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