Rutherford Model (English)

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2-5: Rutherford's Atomic Model

As discussed

in the preceding page,

we should consider that

Thomson's raisin bread model

(plum pudding model)

was unsuccessful,

because it could

not explain the results

of the alpha particle scattering experiment

Geiger and Marsden.

Let us look back

upon the experimental results;

we can summarize them

as follows:

[The Results of Geiger and Marsden's experiment]

	(1)	Almost all the incident alpha particles go straight and are scarcely scattered.
	(2)	Only occasionally such a large-angle scattering through an angle greater than 90 degrees or near 180 degrees occurs.
	(3)	The scattering rate (or probability) depends on the atomic weight of the target; the more the atomic weight, the larger the probability.

[The Rutherford Model of the Nuclear Atom]

Considering the results

of Geiger and Marsden's

experiment

and the failure

of Thomson's atomic model,

E. Rutherford

(UK, 1871 - 1937)

proposed a model

in which

the electric charge +Ze

in an atom

is not distributed

over the whole area

of the atom

but concentrates

in a small area

(1911).

He thought as follows:

The charge +Ze

is localized to be

a cluster or a group

and the alpha particle

is scattered by

Coulomb's repulsive force,

(C. A. de Coulomb:

France, 1736 - 1806).

Namely, his idea is that

the large-angle scattering

of the alpha particle

is brought about

by a single scattering

due to Coulomb's repulsive force

between the charge

of the alpha particle,

+2e, and

that of the cluster,

+Ze.

The cluster is called

atomic nucleus

or simply nucleus.

Rutherford's atomic model

is often called

Rutherford model

or sometimes

Rutherford model of the nuclear atom.

A schematic image

of the Rutherford model

is shown in the following

figure,

where a black big ball

at the center

is the nucleus and

small red points moving

around the nucleus

are electrons.

The Rutherford Model of the Nuclear Atom

A black big ball

at the center

is the nucleus and

small red points moving

around the nucleus

are electrons.

In 1903,

H. Nagaoka

(Japan, 1865 - 1950)

proposed a ``saturnian model'',

which was referred to

in Rutherford's paper.

You should note

that the Nagaoka's model

had been proposed

several years

before Rutherford's model.

[Rutherford Scattering]

Rutherford investigated

whether the results

of the experiment

of the alpha particle scattering

could be explained well

by the Rutherford model

of the nuclear atom.

He derived the famous

Rutherford scattering formula

standing on the viewpoint

of the Rutherford model

and he found

that the results

of this formula

fit well to

the experimental data

(1911).

Thus the firm position

of the Rutherford model

was established.

Rutherford assumed

that the total positive

charge in an atom, +Ze,

concentrates on the central

point of the atom,

i.e., the nucleus,

and the incident alpha particle

is scattered

with a repulsive Coulomb force exerted

by this nuclear point charge.

(See the following

figure.)

This type of scattering

by the Coulomb force

is usually called

Coulomb scattering or

Rutherford scattering.

(See the above

figure.)

The detailed formulation

on Rutherford scattering

will be given

in the following other page.

If you feel it difficult

to understand,

skip it,

and accept the final

Rutherford scattering formula.

2-5-A: Rutherford Scattering

As shown in

the above figure,

the distance between

the incident line

and the nucleus

is the impact parameter

If b = 0,

it is head-on collision,

where the incident alpha particle

would recoil

in the direction

of 180 degrees.

If b

is small,

the trajectory

would be curved largely.

If b is large

and the particle

goes along a path

far from the nucleus,

then the repulsive force

exerted by the nucleus

is weak and

the trajectory would not

curve so much.

Namely,

the trajectory is determined by the impact parameter only

for a constant incident speed

v₀,

This feature are shown

in the following graph

which is drawn

by using the numerical data

obtained from the solution

discussed on the page

(2-5-A).

[Angular Distribution of Rutherford Scattering]

As seen in the above figure,

if the incident line

is close to the nucleus

of the target atom

or if the impact parameter

is small,

the trajectory

of the alpha particle

would be largely

curved and

it would be scattered

backward.

In contrast,

if the impact parameter

is large,

then the alpha particle

goes through far

from the nucleus and

the trajectory

does not curve so much.

In the realistic experiment,

the number of the incident

alpha particles

in a unit area

is almost constant

independently of the

impact parameter.

The number of the

incident particles

in a unit area

and a unit time,

is called the

intensity

of the incident alpha rays.

We may consider that

the intensity N

of the incident alpha rays

is constant.

It is called

the angular distribution

what rate among

the many incident particles

is scattered in the direction

of a certain angle.

Strictly speaking,

the angular distribution

differential scattering cross section

is defined by dividing

the number of

the particles scattered

in a unit solid angle

of a certain scattering angle

in a unit time

by the intensity N.

Rutherford obtained

the differential scattering

cross section

on the basis of

Newtonian mechanics

(1911).

The details of deriving

the cross section

are explained on the page

2-5-A

The result is shown

in the following formula,

This Rutherford's formula

was well fit to experiment.

Thus the validity

of Rutherford's model

of the nuclear atom

has been established.

As stated above,

Rutherford derived

the angular distribution

of the alpha particle scattering

on the basis

of Newtonian mechanics.

Newtonian mechanics

is sometimes called

classical mechanics,

and it was later

clarified that

Newtonian mechanics

is not always correct

in the microscopic world,

i.e.

the world of

atoms and atomic nuclei.

It has been made clear

some years later that

the correct theory

to describe such

a microscopic world

is quantum mechanics.

It has however been

clarified that,

as long as Rutherford scattering

is concerned,

both classical mechanics

and quantum mechanics

give the same angular distribution.

[The Size of the Charge of Atomic Nucleus]

The angular distribution

derived by

Rutherford was

very successful not only

in the angular dependence

but in a by-product.

As seen in the above

Rutherford formula,

contains a factor Z.

By measuring

this scattering

probability precisely,

we can obtain

the value of Z.

The quantity Z

is the number of the electrons

surrounding the nucleus in an atom.

Before Rutherford's formula,

the value of Z

had only been known

to be about a half

of the atomic weight.

However, precise measurements

of the angular distribution

of Rutherford scattering

have shown the precise value

of the charge

of the atomic nucleus

and also the number

of electrons in an atom.

Consequently,

it was made clear

that Z is exactly equal to the atomic number.

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