Transcript Today’s Menu • • • • Why study nuclear physics Why nuclear physics is difficult Course synopsis. Notation & Units Tony Weidberg Nuclear Physics Lectures.

Today’s Menu
•
•
•
•
Why study nuclear physics
Why nuclear physics is difficult
Course synopsis.
Notation & Units
Tony Weidberg
Nuclear Physics Lectures
1
What is the use of lectures
• Definition of a lecture: 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.
• Conclusion: to make lectures useful YOU
have to participate, ask questions ! If you
don’t understand something the chances are
>50% of the audience doesn’t as well, so
don’t be shy !
Tony Weidberg
Nuclear Physics Lectures
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 
experimental nuclear astrophysics is a
hot topic.
Tony Weidberg
Nuclear Physics Lectures
3
Practical Applications
• Nuclear fission for energy generation.
– No greenhouse gasses
– Safety and storage of radioactive material.
• Nuclear fusion
– No safety issue (not a bomb)
– Less radioactive material but still some.
• Nuclear transmutation of radioactive waste
with neutrons.
– Turn long lived isotopes  stable or short lived.
• Every physicist should have an informed
opinion on these important issues!
Tony Weidberg
Nuclear Physics Lectures
4
Medical Applications
• Radiotherapy for cancer
– Kill cancer cells.
– Used for 100 years but can be improved by better
delivery and dosimetery
– Heavy ion beams can give more localised energy
deposition.
• Medical Imaging
–
–
–
–
MRI (Nuclear magnetic resonance)
X-rays (better detectors  lower doses)
PET
Many others…see Medical & Environmental short
option.
Tony Weidberg
Nuclear Physics Lectures
5
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
– Forenesic (eg date As in hair).
– Biology (eg elements in blood cells)
– Archaeology (eg provenance via isotope
ratios).
Tony Weidberg
Nuclear Physics Lectures
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Tony Weidberg
Nuclear Physics Lectures
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Why is Nuclear Physics Hard?
• QCD theory of strong interactions  just
solve the equations …
• At short distance/large Q coupling
constant small  perturbation theory ok
but long distance/small Q, q  large
1 
L  [i     m  ] 
F F  (q    ).A
16

F    A   A  2q( A xA )
Tony Weidberg
Nuclear Physics Lectures
8
Nuclear Physics Models
• Progress with understanding nuclear
physics from QCD=0  use simple,
approximate, phenomenological models.
• Liquid Drop Model: phenomenology + QM +
EM.
• Shell Model: look at quantum states of
individual nucleons  understand
spin/parity magnetic moments and
deviations from SEMF for binding energy.
Tony Weidberg
Nuclear Physics Lectures
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Course Synopsis - 1
• Liquid Drop Model and SEMF.
• Applications of SEMF
– Valley of stability.
– ab decays.
– Fission & fusion.
• Limits of validity of liquid drop model
(shell model effects)
Tony Weidberg
Nuclear Physics Lectures
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Course Synopsis - 2
• Cross Sections
– Experimental definition
– FGR theory
– Rutherford scattering
– Breit-Wigner resonances
• Theory of ab decays.
• Particle interactions in matter
– Simple detectors for nuclear/particle
physics.
Tony Weidberg
Nuclear Physics Lectures
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Corrections
• To err is human … and this is a new
course  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!).
Tony Weidberg
Nuclear Physics Lectures
12
The Minister of Science
• This is a true story honest.
• Once upon a time the government science
minister visited the Rutherford Lab (UK
national lab) and after a days visit of the
lab was discussing his visit with the lab
director and he said …
• I hope that you all have a slightly better
grasp of the subject by the end!
Tony Weidberg
Nuclear Physics Lectures
13
Notation
• Nuclei are labelled ZA El where El is the
chemical symbol of the element, mass
number A = number of neutrons N + number
of protons Z. eg 37 Li
• Excited states labelled by * or m if they are
metastable (long lived).
Tony Weidberg
Nuclear Physics Lectures
14
Units
•
SI units are fine for macroscopic objects like
footballs but are very inconvenient for nuclei and
particles  use natural units.
Energy: 1 eV = energy gained by electron in being
accelerated by 1V.
•
–
•
Mass: MeV/c2 (or GeV/c2)
–
–
•
1 eV/c2 = e/c2 kg.
Or use AMU defined by mass of 12C= 12 u
Momentum: MeV/c (or GeV/c)
–
•
1 eV/c = e/c kg m s-1
Cross sections: (as big as a barn door)
–
•
1 eV= e J.
1 barn =10-28 m2
Length: fermi 1 fm = 10-15 m.
Tony Weidberg
Nuclear Physics Lectures
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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
Tony Weidberg
Nuclear Physics Lectures
16
Nuclear Mass Measurements
• Measure relative masses by energy
released in decays or reactions.
– X  Y +Z + DE
– Mass difference between X and Y+Z is
DE/c2.
• Absolute mass by mass spectrometers
(next transparency).
• Mass and Binding energy:
• B = [Z MH + N Mn – M(A,Z)]/c2
Tony Weidberg
Nuclear Physics Lectures
17
Mass Spectrometer
• Ion Source
• Velocity selector 
electric and magnetic
forces equal and
opposite
– qE=qvB  v=E/B
• Momentum selector,
circular orbit
satisfies:
– Mv=qBr
– Measurement r
gives M.
Tony Weidberg
Detector
Ion Source
Nuclear Physics Lectures
Velocity
selector
18
Binding Energy vs A
• B increases with A up to 56Fe and then
slowly decreases. Why?
• Lower values and not smooth at small
A.
Tony Weidberg
Nuclear Physics Lectures
19
Nuclear Sizes & Isotope Shift
• Coulomb field modified by finite size of
nucleus.
• Assume a uniform charge distribution in the
Ze
r 3
E


(
)
nucleus. Gauss’s law 
2 R
4 0r
integrate and apply boundary conditions
V (r )  
Zer 2
8 0 R
3

3 Ze
8 0 R
• Difference between actual potential and
Coulomb
2
DV (r )  
Zer
8 0 R
3

3Ze
Ze

8 0 R 4 0r
(r  R )
• UseR 1st order perturbation theory
DE   4r  ( r )[ eDV ( r )] ( r )dr
2
0
Tony Weidberg
*
 ( r )  2(
Z 3/ 2
Z
)
exp( Zr / a0 )  2( )3 / 2
a0
a0
Nuclear Physics Lectures
20
Isotope Shifts
R
Zer 2
3Ze
Ze
DE   4r 4( Z / a ) ( e)[ 


] dr
3
8 0 R 4 0r
8 0R
0
2
3
4R 5
 4 r r dr 
5
0
R
3
4

R
2
 4 r dr 
3
0
R
2 2
R
21
4

r
dr  2R 2

r
0
Ze
4  4  3 
2
DE  ( 4e)( Z / a)
R [      2 ]
4 0
10  3  2 
3
2
2
2Ze R
3
DE 
(Z / a0 )
5 0
Tony Weidberg
Nuclear Physics Lectures
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Isotope Shifts
• Isotope shift for optical spectra
• Isotope shift for X-ray spectra (bigger
effect because electrons closer to
nucleus)
• Isotope shift for X-ray spectra for
muonic atoms. Effect greatly enhanced
because m~ 207 me and a0~1/m.
• All data consistent with R=R0 A1/3 with
R0=1.25fm.
Tony Weidberg
Nuclear Physics Lectures
22
Frequency shift of an optical
transition in Hg at =253.7nm
for different A relative to A=198.
Data obtained by laser
spectroscopy.
The effect is about 1 in 107.
(Note the even/odd structure.)
DE/h (GHz)
Isotope Shift in Optical Spectra
Bonn et al Z Phys A 276, 203
(1976)
A2/3
Tony Weidberg
Nuclear Physics Lectures
23
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)
Tony Weidberg
Nuclear Physics Lectures
24
58Fe
Data on Isotope Shift of K Xrays
from muonic atoms [in which a
muon with m=207me takes the place
of the atomic electron].
56Fe
Because a0 ~ 1/m the effect is
~0.4%, much larger than for an
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.
54Fe
Shera et al Phys Rev C 14, 731
(1976)
Tony Weidberg
Nuclear Physics Lectures
Energy (keV)
25
SEMF
• Aim: phenomenological understanding of
nuclear binding energies as function of A &
Z.
• Nuclear density constant (see lecture 1).
• Model effect of short range attraction due to
strong interaction by liquid drop model.
• Coulomb corrections.
• Fermi gas model  asymmetry term.
• QM pairing term.
• Compare with experiment: success & failure!
Tony Weidberg
Nuclear Physics Lectures
26
Liquid Drop Model Nucleus
• Phenomenological model to understand binding
energies.
• Consider a liquid drop
– Ignore gravity and assume no rotation
– Intermolecular force repulsive at short distances, attractive
at intermediate distances and negligible at large distances
 constant density.
E=-an + 4R2T B=an-bn2/3
• Analogy with nucleus
– Nucleus has constant density
– From nucleon nucleon scattering experiments: Nuclear
force has short range repulsion and attractive at
intermediate distances.
– Assume charge independence of nuclear force, neutrons
and protons have same strong interactions check with
experiment!
Tony Weidberg
Nuclear Physics Lectures
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Mirror Nuclei
• Compare binding energies of mirror nuclei
(nuclei n p). Eg 73Li and 74Be.
• Mass difference due to n/p mass and Coulomb
energy.
R
Q( r )
dQ
0 4 0 r
E
Q( r )  Ze( r / R)3 dQ  3 Zer 2 / R 3
3( Ze)2 r 5
( Ze)2
E
dr  ( 3 / 5)
6
4 0 R
0 4 0 r R
R
3 e2
DEc ( Z , Z  1) 
[ Z ( Z  1)  ( Z  1)(Z  2)] ; Z ~ A / 2 ; R  A1 / 3
5 40 R
DEC ( Z , Z  1)  A2 / 3
Tony Weidberg
Nuclear Physics Lectures
28
nn and pp interaction same
(apart from Coulomb)
“Charge symmetry”
Tony Weidberg
Nuclear Physics Lectures
29
Charge Symmetry and Charge
Independence
• Mirror nuclei showed that strong
interaction is the same for nn and pp.
• What about np ?
• Compare energy levels in “triplets” with
same A, different number of n and p. e.g.
22
10Ne
22
11Na
22
12Mg
• Same energy levels for the same spin
states  SI same for np as nn and pp.
Tony Weidberg
Nuclear Physics Lectures
30
Charge Independence
23
23
11Na
12
Mg
• Is np force is same
as nn and pp?
• Compare energy
levels in nuclei with
same A.
• Same spin/parity
states have same
energy.
• np=nn=pp
22
22 Ne
10
Tony Weidberg
Nuclear Physics Lectures
22
11Na
12Mg
31
Charge Independence of Strong
Interaction
• If we correct for n/p mass difference
and Coulomb interaction, then energy
levels same under n p.
• Conclusion: strong interaction same
for pp, pn and nn if nucleons are in the
same quantum state.
• Beware of Pauli exclusion principle! eg
why do we have bound state of pn but
not pp or nn?
Tony Weidberg
Nuclear Physics Lectures
32
Asymmetry Term
• Neutrons and protons are spin ½ fermions 
obey Pauli exclusion principle.
• If other factors were equal  ground state
would have equal numbers of n & p.
Illustration
Neutron and proton states with same
spacing D.
Crosses represent initially occupied states
in ground state.
If three protons were turned into neutrons
the extra energy required would be 3×3 D.
In general if there are Z-N excess protons
over neutrons the extra energy is
((Z-N)/2)2 D. relative to Z=N.
Tony Weidberg
Nuclear Physics Lectures
33
Asymmetry Term
• From stat. mech. density of states in 6d phase space = 1/h3
dN 
4p 2dpV
h3
• Integrate to get total number of protons Z, & Fermi Energy (all
states filled up to this energy level).
Z  (8 / 3) (pFV / h)
3
PF  ( 3 / 8 )1 / 3
h
R0
1/ 3
• Change variables p  E
dN / dE 
dN / dp
 AE1 / 2
dp / dE
h2
Z
 
 A
E
Z
2/ 3
EF  ( 3 / 8 )
2  A
2mR 0  
2/ 3
F
3/ 2
 E dE
 E  E0
 ( 3 / 5)EF
F
1/ 2
 E dE
0
Tony Weidberg
Nuclear Physics Lectures
34
Asymmetry Term
P
ETotal
2
3
Zh
Z
 ( 3 / 8 )2 / 3
 
2
5
2mR 0  A 
ETotal 
ETotal
K
A
2/ 3
Z
5/ 3

 N5 / 3

2/ 3
y  NZ
KA5 / 3
 2 / 3 (1  y / A)5 / 3  (1  y / A)5 / 3
A

• Binomial expansion keep lowest term in y/A
( N  Z )2
E  K
A
• Correct functional form but too small by factor of 2. Why?
Tony Weidberg
Nuclear Physics Lectures
35
Pairing Term
• Nuclei with even number of
n or even number of p more
tightly bound fig.
• Only 4 stable o-o nuclei cf
153 e-e.
• p and n have different
energy levels  small
overlap of wave functions.
Two p(n) in same level with
opposite values of jz have
AS spin state  sym spatial
w.f. maximum overlap
maximum binding energy
because of short range
attraction.
Tony Weidberg
Neutron separation energy in Ba
Nuclear Physics Lectures
Neutron number
36
Pairing Term
• Phenomenological fit to A dependence
1 / 2
E  A
• Effect smaller for larger A
e-e
e-o
o-o
Tony Weidberg

+ive
0
-ive
Nuclear Physics Lectures
37
Semi Empirical Mass Formula
• Put everything together:
B( N , Z )  aA  bA
2/ 3
( N  Z )2
Z2

c
 d 1/ 3  1/ 2
A
A
A
• Fit to measured binding energy.
–
–
–
–
–
–
Fit not too bad (good to <1%).
Deviations are interesting  shell effects.
Coulomb term agrees with calculation.
Asymmetry term larger ?
Explain valley of stability.
Explains energetics of radioactive decays, fission
and fusion.
Tony Weidberg
Nuclear Physics Lectures
38
The Binding Energy per
9.0
nucleon of beta-stable (odd A)
nuclei.
a
b
15.56
17.23
c
23.285
d
0.697

+12 (o-o)

0 (o-e)

-12 (e-e)
B/A (MeV)
Fit values in MeV
7.5
Tony Weidberg
Nuclear Physics Lectures
39
A
Valley of Stability
• SEMF allows us
to understand
valley of stability.
• Low Z,
asymmetry term
 Z=N
• Higher Z,
Coulomb term 
N>Z.
Tony Weidberg
Nuclear Physics Lectures
40