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 Rutherford Scattering.

6.1 The Atomic Models of Thomson and Rutherford

6.2 Definition of Cross Section6.2 Rutherford Scattering6.3 Structure of the Nucleus

CHAPTER 6Rutherford ScatteringRutherford Scattering

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

Niels Bohr (1885-1962)

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).

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.

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.

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.

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

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

222 218Rn Po He

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.

Experiments of Geiger and Marsden

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

Experiments of Geiger and Marsden

Geiger showed that many 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

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

It can be shown that the maximum momentum transfer to the particle is:

Determine max by letting Δpmax be perpendicular to the direction of motion:

Before After

v2max emp

v

v2max M

m

p

p e too small!

If an particle is scattered by N electrons:

Try multiple scattering from electrons

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

N = the number of atoms across the thin gold layer, t = 6 × 10−7 m:

still too small!

n =

N = t / d

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.

Rutherford’s Atomic Model

Ernest Rutherford (1871-1937)

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 .

Rutherford Scattering

When b is small,

r is small.

the Coulomb force is large.

θ can be large and the particle can be repelled backward.

cot

1 2

2cot

2

kq qb

M v

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

Rutherford Scattering Equation

Scattering rate as a function of angle

2

2 22

1

cos 2 1 cosk

d cz Z

d E

Rutherford scattering experimentSee figures 6.4,6.5,6.6 and 6.7 in text for experimental

results

Exercise 6.3: Derive the Rutherford Scattering formulae

2

2 22

1

cos 2 1 cosk

d cz Z

d E

1 22

cot2

kq qb

M v

Measuring the Size of Nucleus

Rutherford Scattering: See Figure a), No penetration of nucleus, Nucleus behaves like point charge, Coulomb force lawDoes 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

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

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

22m

k

kZer

E

Measuring the Size of NucleusRutherford 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

5 146

2(79)(1.44 . )3 10 3 10

7.7 10m

eV nmr nm m

eV

Measuring the Size of NucleusNuclear size is measured in Fermi or FemtometersLightest atom ~ 1fmHeaviest atom ~ 10 atom

Electron scattering experiments give

1 3(1.2 )R fm AThe 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