• Main points of today’s lecture:– Born approximation for

electron scattering– Coulomb scattering.

• Main points of last lecture:– Radioactive dating– Sequential decay– Relation between width and

lifetime– Collisions and cross sections

Physic 492 Lecture 6

( )

( )

( )

( )

( )

( ) ( ) hvv

hwv

v

h

vv

h

/rqi3

tgt

2

Ruth

int

/rqi32Born

2

2

iintffi

errdeZ

1qF

:Factor Form

qFdd

dd

:case scattering Coulomb

Verd2

mf

:ic)relativist-(nonion approximatBorn order 1st

fdd

:f Amplitude Scattering

EkVk2w

:RuleGolden s'Fermi

⋅−

⋅−

→

∫

∫

ρ⋅=

⋅Ωσ

=Ωσ

⋅π−

=θ

θ=Ωσ

θ

ρπ

=

Recall our discussion of radius determination from Coulomb scattering

What happens when there is no nuclear interaction?

Time dependent perturbation theory: (Fermi’s golden rule)Do we need to derive it?

( )EV2w

: to fromn transitioof rate for the RuleGolden s'Fermi2

iintffi

fi

ρΨΨπ

=

ΨΨ

→h

Assuming the validity of Fermi’s Golden Rule:

• What is ρ(E)?

Relating Fermi’s golden rule to a cross section

The scattering amplitude

• Main points of today’s lecture:– Born approximation for

electron scattering– Coulomb scattering.

– Charge distributions from electron scattering

– Isotope shifts– Muon and pionic atoms

• Main points of last lecture:– Born approximation for

electron scattering

Physic 492 Lecture 7

( )

( ) ( ) hvvv /rqi3

tgt

2

Ruth

errdeZ

1qF

:Factor Form

qFdd

dd

:case scattering Coulomb

⋅−∫ ρ⋅=

⋅Ωσ

=Ωσ

( )

( )

( )

( ) int

/rqi32Born

2

2

iintffi

Verd2

mf

:ic)relativist-(nonion approximatBorn order 1st

fdd

:f Amplitude Scattering

EkVk2w

:RuleGolden s'Fermi

hwv

h

vv

h

⋅−

→

∫ ⋅π−

=θ

θ=Ωσ

θ

ρπ

=

The Cross section in terms of form factor

Homework hint and result

• In the next homework, you will be asked to Show that the factor

is equal to :

• Replace , replace

and then take the limit.) • With this result:

( ) 23

2e

uuqiexpudk

2m

∫⋅−⋅

π

vv

h

RuthddΩσ

( ) ( )∫∫

δ−⋅−⋅⋅−⋅→δ u

uuqiexpudlimby u

uqiexpud 3

0

3 vvvv

The Form Factor

Simple models for nuclear charge distribution

• Assume a sharp spherical charge distribution.

What causes the deep diffraction minima.

• Diffraction occurs due to the interference of different parts of the wavefunction that traverse the nucleus

Real data – extraction of ρ(r).• Why no deep minima?

• What approach works?

• The results:

/d mb srd

qh

Some factual corrections

Other probes of the nuclear charge distributions

• Atomic lines of muonic or electonic atoms:• Shift is due to finite size of nuclei:

2

20

eZm4a

μ

πε=

h

• Main points of today’s lecture:– Isotope shifts– Hadronic scattering– Summary of nuclear sizes

and shapes– Nuclei as liquid drops-Semi-

empirical mass formula• Bulk• Surface• Symmetry• Coulomb

• Main points of last lecture:– Charge distributions from

electron scattering– Isotope shifts

Physic 492 Lecture 8

Muonic atom case

• Wavefunctions:

• Bohr radius is small:

• l value governs overlap and ΔE.

2

20

eZm4a :radiusBohr

μ

πε=

h

an interesting result

• One interesting trend is the dependence of the proton radii uponneutron number of a range of isotopes of the same element.

Hadronic scattering

• What happens when you scatter 14 MeV neutrons on 58Ni?

• Actual approach

Summary

Nuclear Masses and Binding Energies

• Definition of Binding energy

• Properties of nuclear binding energies– Average binding energy is approximately 8 – 9 MeV.

Liquid drop formula

• Bulk term.

• Surface term.