Slide 1

CHAPTER 6

Rutherford Scattering
6.1 The Atomic Models of Thomson and
Rutherford
6.2 Definition of Cross Section
6.2 Rutherford Scattering
6.3 Structure of the Nucleus

Niels Bohr (1885-1962)
The opposite of a correct statement is a false statement. But the opposite of a
profound truth may well be another profound truth.
An expert is a person who has made all the mistakes that can be made in a very
narrow field.

Never express yourself more clearly than you are able to think.
Prediction is very difficult, especially about the future.
- Niels Bohr


Slide 2

Structure of the Atom
Evidence in 1900 indicated that
the atom was not a fundamental unit:
1)

There seemed to be too many kinds
of atoms, each belonging to a distinct chemical
element (way more than earth, air, water, and fire!).

2)

Atoms and electromagnetic phenomena were intimately
related (magnetic materials; insulators vs. conductors;
different emission spectra).

3)

Elements combine with some elements but not with
others, a characteristic that hinted at an internal atomic
structure (valence).

4)

The discoveries of radioactivity, x rays, and the electron
(all seemed to involve atoms breaking apart in some way).


Slide 3

Knowledge of atoms in 1900

Electrons (discovered in
1897) carried the negative
charge.
Electrons were very light,
even compared to the atom.
Protons had not yet been
discovered, but clearly
positive charge had to be
present to achieve charge
neutrality.


Slide 4

Thomson’s
Atomic Model
Thomson’s “plum-pudding”
model of the atom had the
positive charges spread
uniformly throughout a
sphere the size of the atom,
with electrons embedded in
the uniform background.

In Thomson’s view, when the atom was heated, the electrons could
vibrate about their equilibrium positions, thus producing
electromagnetic radiation.
Unfortunately, Thomson couldn’t explain spectra with this model.


Slide 5

Alpha (α) Particle
Scattering can be produce by any type of particle, however
the particle must have small wavelength for good diffraction
and resolution.
Alpha particle is such a particle. It is produce in a radioactive
decay of proton
222

Rn 

218

Po  He



He++ is a ionized helium nucleus (q=+2e) and is called the
alpha (α) particle


Slide 6

Scattering with Alpha (α) Particle
Exercise 6-1 Show that when α particles scatter from an
atom, the scattering angle is inversely proportional to the
distance for closest approach.


Slide 7

Experiments of Geiger and Marsden
Geiger, and Marsden conceived a new technique for
investigating the structure of matter by scattering a particles from
atoms.


Slide 8

Experiments of Geiger and Marsden
Geiger showed that many a particles were scattered from thin
gold-leaf targets at backward angles greater than 90°.

Large scattering angles mean the target is more massive than a
projectile


Slide 9

Electrons can’t back-scatter a particles.
Exercise 6-2 Calculate the maximum scattering angle
corresponding to the maximum momentum change.
Before
After

It can be shown that the maximum
 p max  2 m e v a
momentum transfer to the a particle is:
Determine qmax by letting
Δpmax be perpendicular to
the direction of motion:

q max 

 pa
pa



2me va
M a va



too small!


Slide 10

Try multiple scattering from electrons
If an a particle is scattered by N electrons:
N = the number of atoms across the thin gold layer, t = 6 × 10−7 m:
n=

The distance between atoms, d = n-1/3, is:

N=t/d
still too small!


Slide 11

Rutherford’s Atomic Model
even if the α
particle is scattered from all
79 electrons in each atom
of gold.
Experimental results were
not consistent with
Thomson’s atomic model.
Rutherford proposed that an
atom has a positively
charged core (nucleus)
surrounded by the negative
electrons.

Geiger and Marsden
confirmed the idea in 1913.

Ernest Rutherford
(1871-1937)


Slide 12

Rutherford Scattering
Scattering experiments
help us study matter too
small to be observed
directly.
There’s a relationship
between the impact
parameter b and the
scattering angle q.
When b is small,
r is small.
the Coulomb force is large.
θ can be large and the particle can be
repelled backward.

q 
b
cot  
2
Mav
2
kq1 q 2

cot(q/2)

0

p

q


Slide 13

Rutherford Scattering Equation

In actual
experiments, a
detector is
positioned from θ
to θ + dθ that
corresponds to
incident particles
between b and b +
db.

Scattering rate as a
function of angle

 c

z Z a

d cos q
2
 Ek 
d

p

2

2

2

1

(1  cos q

)

2


Slide 14

Rutherford scattering experiment
See figures 6.4,6.5,6.6 and 6.7 in text for experimental
results
Exercise 6.3: Derive the Rutherford Scattering formulae

q 
b
cot  
2
Mav
2
kq1 q 2

 c

z Z a

d cos q
2
E
 k 
d

p

2

2

2

1

(1  cos q )

2


Slide 15

Measuring the Size of Nucleus
Rutherford Scattering: See Figure a), No penetration of nucleus, Nucleus
behaves like point charge, Coulomb force law
Does not imply that nucleus is a point charge
Force law is still
correct even if the
nucleus was ball of
radius R as long as
the alpha particle
does not penetrate
the nucleus
Alpha particle
penetration:
Rutherford
scattering does not
hold


Slide 16

Measuring the Size of Nucleus
Modification is required to account for charge behind the alpha
particle as it penetrates the nucleus


Slide 17

Size of Nucleus
Exercise 6.4: For a head on collision of an alpha particle
with a nucleus show that the distance of closest approach is

rm 

2 kZ e
Ek

2


Slide 18

Measuring the Size of Nucleus
Rutherford scattering formula can be used to find the size of the
nucleus
Increase the energy of the incoming α particle, the distance of
closest approach will be smaller. At some rm (penetration) the
results from scattering experiment will not agree with Rutherford’s
prediction and that rm with give the nuclear size.
Example: For a alpha particle of 7.7 MeV, the radius of the gold
nucleus is

rm 

2(79)(1.44 eV .nm )
7.7  10 eV
6

5

 3  10 nm  3  10

 14

m


Slide 19

Measuring the Size of Nucleus
Nuclear size is measured in Fermi or Femtometers
Lightest atom ~ 1fm
Heaviest atom ~ 10 atom
Electron scattering experiments give

R  (1.2 fm ) A

13

The nucleus is made up of closely packed spheres of protons and
neutrons
Experiments with 1 GeV electrons hitting the nucleus reveal that
there is appreciable deviation from Rutherford scattering cross
section, showing that neutron and proton’s are not point like but
finite size.
Measurement on size of proton and neutron ~ 1fm