Liquid Drop Model (English)

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2-4: The Liquid Drop Model

Needless to say,

a nucleus is a quantum mechanical many-body system

consisting of many nucleons.

We can not directly watch it

with our eyes.

In such case,

how can we see the structure

of a nucleus?

Of course, the fundamental

theory to govern the nuclear structure

is Quantum Mechnics,

and the basic

equation to describe it

must be the

Schroedinger equation.

It is however very difficult and

almost impossible to

precisely solve

the many-body Schroedinger equation for such a

complicated system as nuclei.

For such a complicated

system, we sometimes assume

a simple model

which is very easy

to treat and understand.

If this model can reproduce

a lot of experimental data

on nuclei in a unified way,

then we could

understand the structure of

the nuclei.

Here, let us discuss

the liquid drop model,

which was the oldest one

introduced for the nuclear

structure.

[The Liquid Drop Model]

Just after the discovery

of the neutron by Chadwick,

Ivanenko and Heisenberg proposed

an idea that nuclei

are composed of protons

and neutrons (1932).

Then it becomes the next question

what structure do nuclei have.

C. Fv. Weizsaecker

(Germany, 1912- ) imaged that

the nucleus might be something

like a raindrop.

This is called a

liquid drop model.

[Weizsaecker-Bethe's Mass Formula]

On the basis of the

liquid drop model,

Weizsaecker and

H. A. Bethe

(Germany, USA, 1906-2005)

proposed a simple formula

for the nuclear binding energy.

Let the proton and neutron

masses be

and , respectively.

And let the mass of nucleus

of the proton number Z

and the neutron number N

M (Z, N

As learned on the the last page,

the binding energy

of a nucleus,

B (Z, N ),

is given by

Weizsaecker and Bethe

found that the above

binding energy

may approximately be obtained by

a formula,

This is the famous

Weizsaecker-Bethe's Mass Formula,

which is sometimes called the

semiempirical mass formula.

Meaning of each term in the above

mass formula

can be understood

according to the idea of

the liquid drop model.

The first term

is proportional to the nuclear volume

and called the volume energy,

which is the main part of

the nuclear binding energy.

The second term is proportional

to the area of the nuclear surface,

and it is considered as the

surface energy

due to the surface tension.

The third term is

the Coulomb energy

which is the energy loss

due to the Coulomb repulsive forces

between protons.

Among the nuclei with

the same mass number A

, the energy loss due to the

Coulomb forces becomes larger

in nuclei

with larger proton number Z.

Because of this Coulomb energy,

nuclei with

larger N compared to

have less energy loss,

so that, in the larger

mass number region,

the neutron number N

is, in general,

greater than the proton number

The next term, the fourth term,

shows an opposite effect;

namely, there is a property

that, the less the difference

between the proton number

and the neutron number

is,

the greater the energy gain becomes.

This effect is represented by the fourth term,

which is called the

symmetry energy.

This effect comes from

the symmetry property

in the nuclear force.

In the nucleus,

a pair of equivalent nucleons

(proton-proton or neutron-neutron)

couples with each other

more strongly than

a pair of non-equivalent nucleons

(proton-neutron).

This is also due to the

property of the nuclear force.

The last term, the fifth term,

represents this effect,

which is called the

even-odd mass difference.

The more detailed discussions

about the last two terms,

the symmetry energy and the

even-odd mass difference,

are omitted here, because they

would be very specialized

and complicated.

The parameters in the

above Weizsaecker-Bethe mass formula

should be adjusted to the experimental

nuclear masses.

The results are

and

[Comparison with Experimetal Data]

In order to confirm

how well the Weizsaecker-Bethe

mass formula can reproduce

the experimental data of

nuclear binding energies,

the comparison with experiment

is shown

in the following figure,

in which the

red curve

shows the values of the

binding energy per nucleon

obtained by

the Weizsaecker-Bethe formula,

and the black points indicate

the experimental data,

which are equivalent to

those in the figure

at the bottom

of the preceding page.

You should be careful

because the scales of ordinates in these

two figures are somewhat varied

for the sake of convenience.

[Comparison between Mass formula and Experiment]

The black points

indicate the experimental data of the

binding energy per nucleon.

The

red curve

shows the values obtained by

the Weizsaecker-Bethe mass formula.

Be careful because

the scale of the ordinate

is largely expanded.

As seen in the above figure,

the Weizsaecker-Bethe mass formula

can reproduce well the

experimental data

for a wide range of nuclei.

We can therefore conclude that

the liquid drop model is enough valid in nuclei.

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