physic 492 lecture 6 - michigan state universitylynch/lecture_wk3.pdf · • main points of...
TRANSCRIPT
![Page 1: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/1.jpg)
• 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
=
![Page 2: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/2.jpg)
Recall our discussion of radius determination from Coulomb scattering
![Page 3: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/3.jpg)
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
![Page 4: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/4.jpg)
Assuming the validity of Fermi’s Golden Rule:
• What is ρ(E)?
![Page 5: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/5.jpg)
Relating Fermi’s golden rule to a cross section
![Page 6: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/6.jpg)
The scattering amplitude
![Page 7: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/7.jpg)
• 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
=
![Page 8: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/8.jpg)
The Cross section in terms of form factor
![Page 9: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/9.jpg)
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
![Page 10: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/10.jpg)
The Form Factor
![Page 11: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/11.jpg)
Simple models for nuclear charge distribution
• Assume a sharp spherical charge distribution.
![Page 12: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/12.jpg)
What causes the deep diffraction minima.
• Diffraction occurs due to the interference of different parts of the wavefunction that traverse the nucleus
![Page 13: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/13.jpg)
Real data – extraction of ρ(r).• Why no deep minima?
• What approach works?
• The results:
/d mb srd
qh
![Page 14: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/14.jpg)
Some factual corrections
![Page 15: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/15.jpg)
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
![Page 16: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/16.jpg)
• 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
![Page 17: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/17.jpg)
Muonic atom case
• Wavefunctions:
• Bohr radius is small:
• l value governs overlap and ΔE.
2
20
eZm4a :radiusBohr
μ
πε=
h
![Page 18: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/18.jpg)
an interesting result
• One interesting trend is the dependence of the proton radii uponneutron number of a range of isotopes of the same element.
![Page 19: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/19.jpg)
Hadronic scattering
• What happens when you scatter 14 MeV neutrons on 58Ni?
• Actual approach
![Page 20: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/20.jpg)
Summary
![Page 21: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/21.jpg)
Nuclear Masses and Binding Energies
• Definition of Binding energy
• Properties of nuclear binding energies– Average binding energy is approximately 8 – 9 MeV.
![Page 22: Physic 492 Lecture 6 - Michigan State Universitylynch/lecture_wk3.pdf · • Main points of today’s lecture: – Born approximation for electron scattering – Coulomb scattering](https://reader033.vdocuments.mx/reader033/viewer/2022042011/5e723959dfce751f962b8241/html5/thumbnails/22.jpg)
Liquid drop formula
• Bulk term.
• Surface term.