Nov 2006, Lecture 1 Nuclear Physics Lectures, Dr. Armin Reichold 1 Lectures 1 Introduction and Overview Nuclear sizes and isotope shifts

Nov 2006, Lecture 1 Nuclear Physics Lectures, Dr. Armin Reichold 1

Lectures 1

Introduction and OverviewNuclear sizes and isotope

shifts

Nov 2006, Lecture 1 Nuclear Physics Lectures, Dr. Armin Reichold

2

1.0 Overview 1.1 User guide to these lectures 1.2 Why study nuclear physics 1.3 Why nuclear physics is

diff(eren)(icul)t 1.4 Course synopsis 1.5 Notation & Units 1.6 Nuclear Masses and Sizes

Mass measurements Isotope Shifts


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1.1 How to use these lectures Definition of a classical lecture:

A lecture is a process whereby notes are transferred from the pages of a lecturer to the pages of the student without passing through the head of either.

Disadvantages: obvious …

Conclusion: to make lectures useful YOU have to participate

annotate the notes: notes are not a replacement for text book(s!). Without your comments writtend during and after the lectures they

are of very little use to all but the lecturer take your own notes “As if you were never given these pages” exception: might be good to write your notes onto the sides of these

ask questions: If you don’t understand something the chances are >50% of the

audience doesn’t either, so don’t be shy !


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1.1 Corrections To err is human … and I am giving half

of this course for the first time lots of mistakes.

Please tell me about any mistakes you find in the notes (I will donate a bottle of wine to the person who finds the most mistakes!).


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1.2 Why Study Nuclear Physics?

Understand origin of different nuclei Big bang: H, He and Li Stars: elements up to Fe Supernova: heavy elements

We are all made of stardust Need to know nuclear cross sections to

understand nucleosynthesis experimental nuclear astrophysics is a “hot” topic.


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1.2 Energy Applications Nuclear fission

No greenhouse gasses but … Safety and storage of radioactive material.

Nuclear fusion Fewer safety issues (not a bomb) Less radioactive material but still some.

Nuclear transmutation of radioactive waste with neutrons. Turn long lived isotopes into stable or short lived

ones Every physicist should have an informed

opinion on these important issues!


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1.2 Medical Applications Radiotherapy for cancer

Kill cancer cells. Used for 100 years but can be improved by better

delivery and dosimetry Heavy ion beams can give more localised energy

deposition. Medical Imaging

MRI (Magnetic Resonance Imaging) uses nuclear magnetic resonances

X-rays (better detectors lower doses) PET (Positron Emission Tomography) Many others…see Medical & Environmental short

option.


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1.2 Other Applications Radioactive Dating

C14/C12 gives ages for dead plants/animals/people.

Rb/Sr gives age of earth as 4.5 Gyr. Element analysis

Forensic (eg date As in hair). Biology (eg elements in blood cells) Archaeology (eg provenance via isotope

ratios).


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1.3 Why is Nuclear Physics diff(eren)(icul)t?

We have QCD as an exact theory of strong interactions just solve the equations …

That’s fine at short distances << size of proton i.e. at large momentum transfers = collisions with high CM

energies >> mproton (HEP) coupling constant is small (asymptotic freedom) perturbation theory works

But it fails at large distances = O(size of proton) coupling constant becomes big perturbation theory fails we don’t know how to solve the equations

)(2

).(16

1][

xAAqAAF

AqFFmiL

Not on syllabus !

Boo

!


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1.3 Nuclear Physics (Super) Models

Progress with understanding nuclear physics from QCD=0

use simple, approximate, phenomenological models inspired by analogies to other system

Semi Empirical Mass Formula (SEMF) SEMF = Liquid Drop Model + Fermi Gas Model +

phenomenology + QM + EM. Shell Model: look at quantum states of individual

nucleons to understand ground and low lying excited states

spin, parity magnetic moments (not on syllabus) deviations from SEMF predictions for binding energy.


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1.4 Overview of Lectures (I)1. Introduction

Fri. Week 1, Lindemann (L) Why do we study Nuclear Physics What will this course cover Shape and density of the nuclei

2. The Semi Empirical Mass Formula (SEMF) Thu. Week 2, Martin Wood (MW)

The liquid drop model The Fermi Gas Model Experimental verification

3./4./5. Using the SEMF and transition to Shell Model Fri. (L) Week 2 & Thu. (MW), Fri (L) Week 3

The valley of nuclear stability Nuclear decays (, , fission, others) Natural radioactivity The end of SEMF: Evidence of magic numbers The Shell Model

Note: lectures in the Martin Wood lecture theatre starting 12:05 lectures in the Lindemann lecture theatre starting 14:05


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1.4 Overview of Lectures (II)6./7. Crossections

Thu. (MW), Frid (L) Week 4, Experiments, natural units, conventions and definitions Fermi’s Golden Rule Rutherford Scattering Breit-Wigner resonances and partial decay widths

Note: No nuclear physics lectures in week 5 !8./9. Theory of Decays

Thu. & Fri. Week 6, (MW) Tunnelling model of -decay Selection rules and decay rates in -decay Fermi theory of -decay


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1.4 Overview of Lectures (III)10./11. Particle Interactions with Matter

Thu. & Fri. Week 7, (MW) dE/dx by ionisation and the Bethe-Bloch formula (9) Photoeffect, Compton Scattering, Bremsstrahlung, Pair

Production Cherenkov radiation

12./13. Applications of Nuclear Physics Thy. & Fri. Week 8, (MW)

Particle Detectors Fission Reactors Bombs Fusion reactors Radioactive dating (notes only)


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The Minister of Science This is a true story honest. Once upon a time the UK science

minister visited the Rutherford Lab (UK national lab. near Didcot) and after a days visit of the lab was discussing his visit with the lab director and he said …<censored>

Your answer should at least have been as good as “air”!


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1.5 Notation Nuclei are labelled: e.g.

El = chemical symbol of the element Z = number of protons N = number of neutrons A = mass number = N + Z

Excited states labelled by * or m if they are metastable (long lived).

ElAZ Li7

3


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1.5 Units SI units are fine for macroscopic objects like

footballs but are very inconvenient for nuclei and particles use appropriate units.

Energy: 1 MeV = kinetic energy gained by an electron in being accelerated by 1MV.

1 eV= 106 x e/[C] x 1 J = 1.602 x 10-19 J Mass: MeV/c2 (or GeV/c2)

1 MeV/c2 = 106 x e/[C] / c2 x 1kg = 1.783 x 10-30 kg Or use Atomic Mass Unit (AMU or u) defined by:

mass of 12C= 12 u 1 u = 1.661 x 10-27 kg = 0.93 GeV/c2

Momentum: MeV/c (or GeV/c) 1 MeV/c = 106 x e/[C] / c x kg

Length: fermi 1 fm = 10-15 m Cross sections: barn = as big as a barn door (to a

particle physicists) 1 barn = 10-28 m2 = 100 fm2

Note: C = Coulombc = speed of light


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1.6 Nuclear Masses and Sizes Masses and binding energies

Absolute values measured with mass spectrometers.

Relative values from reactions and decays. Nuclear Sizes

Measured with scattering experiments (leave discussion until after we have looked at Rutherford scattering).

Isotope shifts in atomic spectra


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1.6 Nuclear Mass Measurements

Lets collect all the experimental facts first ! Measure relative masses by energy released

in decays or reactions. X Y +Z + E Mass difference between X and Y+Z is E/c2.

Absolute masses measured by mass spectrometers (next transparency).

Relation between Mass and Binding energy: B = [Z MH + N Mn – Matom(A,Z)]/c2 or B’ = [Z Mp + N Mn – Mnucleus(A,Z)]/c2

(neglecting atomic binding energy of electrons)


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1.6 Mass Spectrometer

ion sourcevelocity selector

B

E

B

position se

nsitive

detector

momentumselector

Ion Source (e.g. strong laser takes out electrons) Velocity selector:

for electric and magnetic forces to be equal and opposite need

Momentum selector, circular orbit satisfies:

Measurement of x gives rcurv rcurv and v gives M

x=x(rcurv)


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1.6 Binding Energy per nucleon vs. A

Typical way of representing mass measurements B increases with A up to 56Fe and then slowly

decreases. B is very small and not smooth at small A. Why? See SEMF and Shell Modell.

avg.

bin

din

g E

nerg

y,

B

per

nu

cleon [

MeV

]

Mass Number A

Fe


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1.6 Nuclear Sizes and Isotope Shifts

Measure size of nucleus by the effect of its charge distribution on the energy levels of atomic electrons

Simple point like Coulomb field will be modified by finite size of nucleus.

This should be felt most by electrons close to the nucleus i.e. k-shell & L=0

And should be negligible for electrons with minimal overlap with the nucleus, i.e. L>0 (~r L)

study this assuming Hydrogenic ground state wave functions for the electrons

that’s justified even for large Z atoms since k-shell electron does not see much of “outer” electrons


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usual 1/r2 factor

fraction of charge inside

r

1.6 Nuclear Sizes & Isotope Shifts

Assume a uniform distribution of charge Ze in a spherical nucleus of radius R.

Calculate potential inside nucleus Vinside:

Einside via Gauss’s law:

Vinside by integrating Einside and applying boundary conditions

at r=R to match Vinside to usual 1=r2 potential:

Difference between actual potential and Coulomb

32

0

( )4inside

Ze rE

r R

2

30 0

3( )

8 8inside

Zer ZeV r

R R

2

30 0 0

3( ) ( )

8 8 4

Zer Ze ZeV r r R

R R r


23


Use 1st order perturbation theory to calculate energy shift E:

2 *

0

4 ( )[ ( )] ( )R

E r r e V r r dr

3/ 2 3/ 20

0 0

( ) 2( ) exp( / ) 2( )Z Z

r Zr aa a

result of angle integration

Insert approximate Hydrogenic ground state wave function:

22 3

30 0 00

34 4( / ) ( )[ ]

8 8 4

R Zer Ze ZeE r Z a e dr

R R r

52 2

0

44

5

R Rr r dr

2 2

0

14 2

R

r dr Rr

3

2

0

44

3

R Rr dr


24


2 23

00

2( / )

5

Ze RE Z a

3 2

0

4 4 3( 4 )( / ) [ 2 ]

4 10 3 2

ZeE e Z a R

Note: E is proportional to Z4 and R2 most noticeable effect deep inside large Z nuclei

a0 = 0.5 10-10 m


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1.6 Isotope Shifts Look at transitions from l=1 (no isotope shift) to

l=0 (large isotope shift) Preferably look for transitions at low n. Types of isotope shifts in increasing shift order:

Isotope shift for optical spectra: E = O(eV) Isotope shift for X-ray spectra (bigger effect then optical

because electrons closer to nucleus): E = O(0.1 eV) Isotope shift for X-ray spectra for muonic atoms. Effect

greatly enhanced because m~ 207 me and a0~1/m. E = O(keV)

All data consistent with R=R0 A1/3 using R0=1.25fm.

Nov 2006, Lecture 1 26

Energy shift of an optical transition in Hg at =253.7nm for different A relative to A=198.

Data obtained by Doppler free laser spectroscopy.

The effect is about 1 in 107. (Note the even/odd structure.)

Bonn et al Z Phys A 276, 203 (1976)

1.6 Isotope Shift in Optical Spectra

Need to use higher n wave functions to calculate this Use Zeff ≈ Z-n expect (Zeff/Z)4 dependence in E Why is E ~ A2/3 ? … E ~ R2 (see before) and R=R0*A1/3

A2/3

E (e

V)

0

40

Note the invisibly small error bars

21 eV

Two lines for odd and even A!See SEMF pairing term later


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Data on the isotope shift of K X ray lines in Hg. The effect is about 1 in 106. Again the data show the R2 = A2/3 dependence and the even/odd effect. Lee et al, Phys Rev C 17, 1859 (1978)

A2/3

E (

eV)

0.5

0

1.6 Isotope Shift in X-Ray Spectra

Bigger shifts as expected

Again two lines ~ A2/3

Nov 2006, Lecture 1 28

Data on Isotope Shift of K Xrays from muonic atoms [in which a muon with m=207me takes the place of the atomic electron].

The large peak is 2p3/2 to 1s1/2. The small peak is 2p1/2 to 1s1/2. The size comes from the 2j+1 statistical weight.

Shera et al Phys Rev C 14, 731 (1976)

58Fe

56Fe

54Fe

Energy (keV)

1.6 Isotope Shift in muonic atoms See dependence on Rnucl Because a0 ~ 1/m the

effect is ~0.4%, i.e. much larger than for an electron

Changing Rnucl by increasing A gives changes in isotope shifts of 2 keV 2keV


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1.6 Isotope Shift Conclusions All types of isotopes shifts show ~A2/3 as

expected for an R2nucl dependence

This holds for all types of nuclei When fitting the slopes we find the same

R0 in Rnucl=R0*A1/3

This tells us that the nuclear density is a universal constant

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Nov 2006, Lecture 1 Nuclear Physics Lectures, Dr. Armin Reichold 1 Lectures 1 Introduction and Overview Nuclear sizes and isotope shifts