1. Advanced Nuclear Physics
MOHAMMAD IMRAN AZIZ
Assistant Professor
PHYSICS DEPARTMENT
SHIBLI NATIONAL COLLEGE, AZAMGARH
(India).
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
2. Nuclides
The total number of protons in the nucleus of an atom is called the atomic
number of the atom and is given the symbol Z. The number of electrons in an
electrically-neutral atom is the same as the number of protons in the nucleus.
The number of neutrons in a nucleus is known as the neutron number and is
given the symbol N. The mass number of the nucleus is the total number of
nucleons, that is, protons and neutrons in the nucleus. The mass number is
given the symbol A and can be found by the equation Z + N = A.
.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
3. why electrons cannot exist inside a nucleus:
Nuclear constituents and their properties
This can be explained mathematically also using Heisenberg's
uncertainty principle as follows
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
4. Confinement Calculation
Electtron is not found in the nucleus, it means we are talking about free electron. That is free
electron does not exist.n ....> p + e, here this is beta deacy and electron that becomes free is
emitted out of the nucleus as free electron as can not exist due to Heisenberg uncertainty.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
5. It is common practice to represent the total angular momentum of a
Nuclear Spin
nucleus by the symbol I and to call it "nuclear spin". For electrons in
atoms we make a clear distinction between electron spin and electron
orbital angular momentum, and then combine them to give the total
angular momentum. But nuclei often act as if they are a single entity
with intrinsic angular momentum I. Associated with each nuclear spin
is a nuclear magnetic moment which produces magnetic interactions
with its environment.
The nuclear spins for individual protons and neutrons parallels the
treatment of electron spin, with spin 1/2 and an associated magnetic
moment. The magnetic moment is much smaller than that of the
electron. For the combination neutrons and protons into nuclei, the
situation is more complicated.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
6. • A characteristic of the collection of protons and
neutrons (which are fermions) is that a nucleus of
odd mass number A will have a half-integer spin and
a nucleus of even A will have integer spin. The
suggestion that the angular momenta of nucleons
tend to form pairs is supported by the fact that all
nuclei with even Z and even N have nuclear spin I=0.
The half-integer spins of the odd-A nuclides suggests
that this is the nuclear spin contributed by the odd
neutron
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
7. Nuclear Magnetic Moments
• Associated with each nuclear spin is a magnetic moment
which is associated with the angular momentum of the
nucleus. It is common practice to express these magnetic
moments in terms of the nuclear spin in a manner parallel to
the treatment of the magnetic moments of electron spin and
electron orbital angular momentum.
• For the electron spin and orbital cases, the magnetic
moments are expressed in terms of a unit called a Bohr
magneton which arises naturally in the treatment of
quantized angular momentum
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
8. Electric Quadrupole Moments of
Nuclei
The nuclear electric quadrupole moment is a parameter which
describes the effective shape of the ellipsoid of nuclear charge
distribution. A non-zero quadrupole moment Q indicates that the
charge distribution is not spherically symmetric. By convention, the
value of Q is taken to be positive if the ellipsoid is prolate and
negative if it is oblate.
The quantity Q0 is the classical form of the calculation
represents the departure from spherical symmetry in the rest
frame of the nucleus.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
9. Generally, the measured quantity is proportional to the z-component of the
magnetic moment (the component along the experimentally determined
direction such as the direction of an applied magnetic field, etc. ). In this
treatment, the use of a "gyromagnetic ratio" or "g-factor" is introduced. The g-
factor for orbital is just gL = 1, but the electron spin g-factor is approximately
gS = 2
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
10. For free protons and neutrons with spin I =1/2, the magnetic moments
are of the form
where
Proton: g = 5.5856912 +/- 0.0000022
Neutron: g = -3.8260837 +/- 0.0000018
The proton g-factor is far from the gS = 2 for the electron, and even the uncharged
neutron has a sizable magnetic moment! For the neutron, this suggests that there is
internal structure involving the movement of charged particles, even though the net
charge of the neutron is zero. If g=2 were an expected value for the proton and g=0
were expected for the neutron, then it was noted by early researchers that the the
proton g-factor is 3.6 units above its expected value and the neutron value is 3.8 units
below its expected value. This approximate symmetry was used in trial models of the
magnetic moment, and in retrospect is taken as an indication of the internal structure
of quarks in the standard model of the proton and neutron
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
11. Note that the maximum effective magnetic moment of a nucleus in
nuclear magnetons will be the g-factor multiplied by the nuclear
spin. For a proton with g = 5.5857 the quoted magnetic moment is m
= 2.7928 nuclear magnetons.
Nuclear Magnetic moment
Nuclide
spin I m in mN
n 1/2 -1.9130418
p 1/2 +2.7928456
2H(D) 1 +0.8574376
17O 5/2 -1.89279
57Fe 1/2 +0.09062293
57Co 7/2 +4.733
93Nb 9/2 +6.1705
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
12. Nuclei
• Parameters of nuclei
• Strong Interaction
• Binding Energy
• Stable and Unstable Nuclei
• Liquid-Drop Model
Numerous Applications:
• nuclear power
• applications in medicine, biology and
chemistry
• evolution of stars and the Universe
• nuclear weapons
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
13. Size of Nuclei and Rutherford Scattering
1/3
R R0 A
- depends weakly on A
(number of nucleons in
the nucleus)
R0 1.2 1015 m 1.2 fm Geiger, Marsden,
238 1/3
Rutherford,1910
R 92 U 1.2 fm 238 7.4 fm
Scattering pattern was consistent with that expected for
-particles: bare He nuclei scattering of particles by pointlike objects having a charge
of +79e (the charge of the gold nucleus). This allowed
Rutherford to put an upper limit on the size of the nucleus
(<310-14m for gold).
Calculations were strictly classical. However, because of the Coulomb
interaction between alpha-particles and nucleus, the result
miraculously coincides with the exact quantum-mechanical one
(recall the success of the Bohr model for atoms).
To measure the size of a nucleus, one
has to use more energetic particles
(or electrons, which are more R – the fitting
commonly used these days) that get parameter
pdfMachine
close enough to get inside the nucleus.
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
14. Nuclear Mass
A 12
X C
number of nucleons in the nucleus chemical symbol
6
Z
(mass number of the nucleus) for the element
number of protons in the nucleus 6 protons + 6 neutrons
N A Z
(atomic number of the element)
The neutron number:
Atomic Mass (the mass of a neutral atom):
M Z m p N mn U nucleus / c 2 Z me U el .nucl . / c 2
U nucleus ,U el .nucl . 0 - attraction between nucleons in the nucleus and between
electrons and the nucleus
U el .nucl . U nucleus mp c 2 - we can neglect Uel.-nucl. and introduce a
convenient mass unit:
M 12 C
1u 6
1.66054 1027 kg 931.49MeV / c 2
12
Mass Unit
m p proton mass 1.007277u
mn neutron mass 1.008665u
m electron mass 0.000549u
pdfMachine
e
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
15. Nuclear Density
MA 1au 1au
3 3 3 3 3 3
R R0 A R0
4 4 4
1.66 1027 kg
2.4 1017 kg / m3
3 15 3
1.2 10 m
4
!
The density of neutron stars is comparable with that of
nuclei.
(Unstable) isotopes of tin and zinc.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
16. The Need for a “Strong Force”
Which interaction controls the size of nucleons? This cannot be electromagnetic interaction:
protons have the same electric charge (they would repel each other) and also there are attractive
forces between protons and electrically neutral neutrons.
Strong Interaction:
binds protons to protons, neutrons to neutrons, and protons to neutrons with roughly the
same force
does not affect certain other kinds of particles (specifically electrons)
is short-ranged (the range ~ 2 fm). Nucleons separated by a larger distance exert no strong
forces on each other.
These observations are explained by the quark model of nucleons.
g
Nucleons are the combination of quarks that are strong-interaction-
u
neutral (like an electrically-neutral atom). Two nucleons interact only if
they are close enough that the distances between various pairs of
d
quarks are significantly different.
u
quarks
and
gluons
~ 10-15 m
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
17. Structure of Matter
g
u
El.-mag. interaction
u d
determines the size of
quarks
atoms
and
gluons
~ 10-15 m
~ 10-10 - 10-9 m Protons & Neutrons (nucleons)
are almost an empty space (the
Atom is almost an
quark size is <10-18 m)
empty space (the
nuclear volume is ~10-
15 of the atomic
volume) Strong interaction
determines the size of
nuclei and nucleons
pdfMachine
A pdf writer that produces quality PDF files with -15 ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
~ 10 - 10 -14m
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
18. Binding Energy
Binding energy: Esys EB E parts EB E parts Esys -the binding energy is positive for
a bound system
Recall a H atom: the binding energy is 13.6 eV (the ground state energy with sign “minus”).
n
n p
p
add EB
n n
E parts mi c 2
Esys p
We can compute the binding energy if we know masses of a system and its constituents:
EB mass of all particles consituting the atom mass of the atom c 2
Z m1 H N mn M A X c 2 EB M c 2
1 Z
mass deficit
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
19. Binding Energy curve
Because of the short-range character of strong interaction (basically, between nearest and next-to-
nearest neighbors), the interaction energy per nucleon with increasing Z saturates at the level ~
(Z/2)(# of neighbors).
Binding energy per nucleon (EB/A), MeV
increasing Z is caused by the long-range Coulomb B Z 2
The decrease of the binding energy with
repulsion of protons:
Mass number, A
The binding energy ~ 10MeV/nucleon is ~1% of the nucleon’s rest energy: we can
pdfMachine
consider the nucleus as a system of individual nucleons
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
20. Liquid-Drop Model
2 2
A “semi-classical” model of the
EB aV A as A2/3 ac
Z
aa
A 2Z
A1/3 A
nucleus: describes reasonably well
the dependence EB(A):
dq r
q r
- charge density
q r dq r
4 3 2
r 4 r dr
R0
3
U R0
0
4 0 r
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
21. Liquid-Drop Model (cont’d)
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
22. Limitations of Liquid-Drop Model
Maria Goeppert-Mayer, J.H.D.Jensen
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
23. Stable Nuclei
Isotopes: all nuclei that have the same number of
protons (Z) but different number of neutrons (N).
Since the chemical properties of an atom are
determined by the number of its electrons,
isotopes of the same element have almost
identical chemical properties.
Example: naturally occurring isotopes of oxygen
16 17 19
8 0 8 0 8 0
Related questions:
- What makes unstable nuclei unstable?
- What are the mechanisms by which they
transform themselves into stable nuclei?
-Why do light stable nuclei tend to have N Z?
- Why do heavier nuclei tend to have more
neutrons than protons?
- Why are there no stable nuclei with Z>83?
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
24. What makes unstable nuclei unstable?
Both protons - and neutrons are energy
fermions (they obey the exclusion
principle). Nuclei are two- 0
r
component Fermi systems. Each
nuclear energy level can contain
four particles: two protons (s=½)
12
and two neutrons (s=½).
6 C
The potential experienced by nucleons is a 3D potential well. The
ground-state configuration of the carbon-16 nucleus :
If a nucleus is allowed to decrease its energy by transforming “excessive”
protons (neutrons) into neutrons (protons), it will do it! protons neutrons
The processes responsible for these transformations are driven by weak interaction (the fourth
fundamental interaction):
The weak interaction (unlike the strong interaction) affects both quarks and leptons, (unlike the
el.-mag. interaction) can affect electrically neutral particles, and (unlike gravity) does not affect
photons.
The effective range of the weak interaction is ~ 10-18m.
Some important transformation processes driven by weak interaction:
n e p pdfMachine p n e
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in p e n
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
25. Why N Z for light nuclei
13 13
8 O energy 7 N energy
If the electrostatic repulsion
t1/2 8.9ms of protons can be neglected
(this is the case of light
nuclei: recall that the
positive electrostatic energy
Z2), the nucleus tends to
keep approximately equal
numbers of protons and
neutrons.
protons neutrons protons neutrons
14 14
6 C energy 7 N energy
Even in this case, the nucleus
t1/2 5730 y
can still lower its total
energy: the rest energy of
neutron is slightly more than
the rest energy of a proton
and an electron.
pdfMachine
A pdf writer that produces quality PDF files with ease!
protons neutrons protons
aziz_muhd33@yahoo.co.in neutrons
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
26. Why N > Z for heavy nuclei
44 44
22 Ti energy 20 Ca energy In the heavy nuclei, the electrostatic
energy cannot be neglected. As a
result, the protons’ energy levels are
“pushed up” with respect to the
neutrons’ levels. In the “otherwise
stable” 44Ti, two protons undergo
the transformation into neutrons,
the end product is stable 44Ca.
. .
. . . .
. . . .
protons neutrons protons neutrons
The proton-neutron disbalance becomes more
pronounced with increasing Z.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
27. 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
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
28. 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
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
29. Mass Spectrometer
• Ion Source
• Velocity selector Detector
electric and magnetic
forces equal and
opposite Velocity
– qE=qvB v=E/B Ion Source
selector
• Momentum selector,
circular orbit satisfies:
– Mv=qBr
– Measurement r gives
M.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
30. Binding Energy vs A
• B increases with A up to 56Fe and then slowly
decreases. Why?
• Lower values and not smooth at small A.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
31. Nuclear Sizes & Isotope Shift
• Coulomb field modified by finite size of nucleus.
• Assume a uniform charge distribution in the nucleus.
Gauss’s law
r 3Ze
integrate and apply boundary conditions
E ( )
2 R
4 0 r
• Difference between actual potential and Coulomb
Zer 2 3Ze
V (r ) 3
8 0 R 8 0 R
• Use 1st order perturbation theory
Zer 2 3Ze Ze
V ( r ) 3
(r R )
8 0 R 8 0 R 4 0r
R Z 3/ 2 Z
2 *
E 4r ( r )[ eV ( r )] ( r )dr ( r ) 2( ) exp( Zr / a0 ) 2( )3 / 2
0
a0 a0
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
32. Isotope Shifts
R
2 3Ze 3 Ze Zer 2
E 4r 4( Z / a ) ( e)[ 3
] dr
8 0R 8 0 R 4 0r
0
R 4R 5 R 4R 3 R 1
4 r 2 r 2dr 2
4 r dr
2
4 r dr 2R
2
0 5 0 3 0 r
Ze 2 4 4 3 3
E ( 4e)( Z / a ) R [ 2 ]
4 0 10 3 2
2 2
2 Ze R 3
E ( Z / a0 )
5 0
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
33. 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 mm~
207 me and a0~1/m.
• All data consistent with R=R0 A1/3 with
R0=1.25fm.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
34. 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 + 4pR2T 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!
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
35. Mirror Nuclei
• Compare binding energies of mirror nuclei (nuclei n
p). Eg 73Li and 74Be.
• Mass difference due to n/p mass and Coulomb
R
energy.
Q( r )
E dQ
0 4 0 r
Q( r ) Ze( r / R ) 3 dQ 3 Zer 2 / R 3
3( Ze ) 2 r 5
R
( Ze ) 2
E 6
dr ( 3 / 5)
0 4 0 r R 4 0 R
3 e2
E c ( Z , Z 1) [ Z ( Z 1) ( Z 1)( Z 2)] ; Z ~ A / 2 ; R A1 / 3
5 4 0 R
EC ( Z , Z 1) A2 / 3
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
36. 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.
– n=number of molecules, T=surface tension, B=binding energy
E=total energy of the drop, a,b=free constants
E=-an + 4pR2T B=an-bn2/3
surface area ~ n2/3
Analogy with nucleus
Nucleus has constant density
From nucleon-nucleon scattering experiments we know:
Nuclear force has short range repulsion and is attractive at intermediate distances.
Assume charge independence of nuclear force, neutrons and protons have
same strong interactions check with experiment (Mirror Nuclei!)
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
37. Coulomb Term
• The nucleus is electrically charged with total charge Ze
• Assume that the charge distribution is spherical and compute the
reduction in binding energy due to the Coulomb interaction
Ze
Q(r )
ECoulomb dQ Q (r ) Ze(r / R )3 dQ 3Zer 2 / R 3 dr
0
4 0 r
to change the integral to dr ; R=outer radius of nucleus
R
3( Ze) 2 r 5 ( Ze) 2
ECoulomb dr (3 / 5)
includes self interaction of last
6
4 0 r R 4 0 R
proton with itself. To correct this
0 replace Z2 with Z*(Z-1)
… and remember R=R0A-1/3
in principle you could take d from this calculation
Z *( Z 1)
BCoulomb ( Z , A) d
but it is more accurate to take it from the overall
1/ 3
A
fit of the SEMF to data (nuclei not totally
pdfMachine
spherical or homogeneous)
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
38. Mirror Nuclei
• Does the assumption of the drop model of constant binding energy for every
constituent of the drop acatually hold for nuclei?
• Compare binding energies of mirror nuclei (nuclei with np). Eg 73Li and
7 Be.
4
• If the assumption holds the mass difference should be due to n/p mass
difference and Coulomb energy alone.
• Let’s compute the Coulomb energy correction from results on previous page
3 e2 3 e2
Ecoulomb ( Z , Z 1) [ Z ( Z 1) ( Z 1)( Z 2)] 2( Z 1)
5 4 0 R 5 4 0 R
Z ~ A / 2 ; R R0 A1/ 3 to find that EC ( Z , Z 1) A2 / 3
Now lets measure mirror nuclei masse, assume that the model holds and
derive DECoulomb from the measurement.
This should show an A2/3 dependence
And the scaling factor should yield the correct R0 of 1.2 fm
pdfMachine
if the assumptions were right
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
39. “Charge
symmetry”
nn and pp
interaction same
(apart from
Coulomb)
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
40. More charge symmetry
• Energy Levels of two mirror nuclei for a number of excited states
• Corrected for n/p mass difference and Coulomb Energy
DEcorrected
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
41. From Charge Symmetry to 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 22 22
10 Ne 11 Na 12 Mg
• If we find the same energy levels for the same spin states
Strong interaction is the same for np as nn and pp.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
42. Charge Independence
DEcorrected
• Same spin/parity states
should have the same
energy.
• Yes: np=nn=pp
• Note: Far more states in
22 Na. Why?
11
• Because it has more np pairs
then the others
• np pairs can be in any Spin-
Space configuration
• pp or nn pairs are excluded
from the totally symmetric
ones by Herr Pauli
22
10Ne
• Note also that 2211Na has the
22
lowest (most bound) state, 12Mg
remember for the deuteron
22
11Na
on next page
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
43. Charge Independence
• We have shown by measurement that:
– If we correct for n/p mass difference and Coulomb interaction, then
energy levels in nuclei are unchanged under n p
– and we must change nothing else! I.e. spin and space wavefunctions
must remain the same!
• Conclusion: strong two-body interaction same for pp, pn and
nn if nucleons are in the same quantum state.
• Beware of the Pauli exclusion principle! eg why do we have
bound state of pn but not pp or nn?
– because the strong force is spin dependent and the most strongly
bound spin-space configurations (deuteron) are not available to nn or
pp. It’s Herr Pauli again!
– Just like 2211Na on the previous triplet level schema
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
44. Volume and Surface Term
• We now have all we need to trust that we can apply the
liquid drop model to a nucleus
– constant density
– same binding energy for all constituents
• Volume term: BVolume ( A) aA
2/3
• Surface term: BSurface ( A) bA
• Since we are building a phenomenological model in which
the coefficients a and b will be determined by a fit to
measured nuclear binding energies we must inlcude any
further terms we may find with the same A dependence
together with the above
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
45. Asymmetry Term
• Neutrons and protons are spin ½ fermions obey
Pauli exclusion principle.
• If all other factors were equal nuclear ground state
would have equal numbers of n & p.
neutrons protons Illustration
n and p states with same spacing .
Crosses represent initially occupied states in ground
state.
If three protons were turned into neutrons
the extra energy required would be 3×3 .
In general if there are Z-N excess protons over
neutrons the extra energy is ((Z-N)/2)2 . relative
to Z=N.
But how big is D ?
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
46. Asymmetry Term
• Assume:
– p and n form two independent, non-interacting
gases occupying their own square Fermi wells
– kT << D
– so we can neglect kT and assume T=0
– This ought to be obvious as nuclei don’t suddenly
change state on a warm summers day!
– Nucleons move non-relativistically (check later if
this makes sense)
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
47. Asymmetry Term
• From stat. mech. density of states in 6d phase space = 1/h3
4 p 2 dpV
dN particle
h3
here N could be the number of protons or neutrons
particle
• Integrate up to pf to get total number of protons Z (or Neutrons N), & Fermi
Energy (all states filled up to this energy level).
3 3 4 3
Z (4 V / 3h ) pF and V R0 A
3
1/ 3 2 2/3
h Z 2 h Z
PF (3 / 4 ) 2/3
and E p EF (3 / 4 ) 4/3
R0 A 2m 2mR02 A
EF
• Change variables p E to find avg. E
E 3/ 2 dE
dN / dp
dN / dE const E1/ 2 E 0
(3 / 5) EF
EF
dp / dE
E1/ 2 dE
0
pdfMachine
These are all standard stat. mech. results!
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
48. Asymmetry Term
Compute total energy of all protons by Z*<E> call this K
2 2/3
P P 3 4/3 h Z
E Total Z E (3 / 4 ) 2
Z
5 2mR0 A
Use the above to compute total energy of Z protons and N neutrons
K
ETotal Z 5/ 3 N 5/ 3 change variables from (Z,N,A)
A2 / 3 to (y,A) with y=N-Z
KA5/ 3
ETotal 2 / 3 5 3 (1 y / A)5/ 3 (1 y / A)5/ 3 where y/A is a small number (e)
A 2
KA
ETotal 5 3 (1 )5/ 3 (1 )5/ 3
2
• Binomial expansion keep lowest term in y/A
2 3 21 35 ( N Z ) 2
ETotal 2 KA K
note! linear terms cancel
9 A ( N Z )2
ETotal ( Fermi Gas ) const *
This terms is only proportional to volume A
comes from a fit of the
(A). It has already been captured by the
Volume term of the liquid pdfMachine
SEMF to measurements
A pdf writer that produces quality PDF files with ease!
drop model aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
analytical ≈ 24 MeV
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
49. Asymmetry term
• From the Fermi Gas model we learn that
– due to the fermionic nature of p and n we loose in
binding energy if the nucleus deviates from N=Z
( N Z )2
• The Asymmetry term: B Asymmetry ( N , Z ) c A
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
50. • Observations:
• Nuclei with even number of n or Pairing Term
even number of p more tightly
Neutron separation energy
bound then with odd numbers.
[MeV] in Ba isotopes
See figure
• Only 4 stable o-o nuclei but 153
stable e-e nuclei.
• p energy levels are Coulomb 56+N
56Ba
shifted wrt n small overlap of
wave functions between n and
p.
• Two p or two n in same energy
level with opposite values of jz
have AS spin state
forced into sym spatial w.f.
maximum overlap
maximum binding energy
Neutron number
because of short range
attraction.
Note: this only holds for nn and pp, not for
np. We don’t have a preference for even A
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
51. Pairing Term
• Measure that the Pairing effect smaller for larger A
• Phenomenological*) fit to A dependence gives A-1/2
BPairing ( A) 1/ 2
A
d
Note: If you want to plot binding energies
e-e +ive versus A it is often best to use odd A only
as for these the pairing term does not
e-o 0 appear
o-o -ive *) For an even more insightful
explanation of the A dependence
read the book by Jelley
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
52. Semi Empirical Mass Formula
• Put everything together:
2 2
2/3 (N Z ) Z
B( N , Z ) aA bA c d 1/ 3 1/ 2
A A A
Volume Surface Asymmetry Coulomb Pairing
Term Term Term Term Term
Lets see how all of these assumptions fit reality
And find out what the constants are
Note: we went back to the simpler Z instead of Z*(Z-1)
2
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
53. Semi Empirical Mass Formula
Binding Energy vs. A for beta-stable odd-A nuclei
Fit parameters
Iron in MeV
a 15.56
Not smooth because Z b 17.23
not smooth function of A
c 23.285
d 0.697
d +12 (o-o)
d 0 (o-e)
d -12 (e-e)
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
54. Semi Empirical Mass Formula
• Conclusions
– Only makes sense for A≥20
– Good fit for large A (good to <1%) in most places.
– Deviations are interesting shell effects.
– Coulomb term constant agrees with calculation.
– Explains the valley of stability (see next lecture).
– Explains energetics of radioactive decays, fission and fusion.
pdfMachine
A pdf writer that produces quality PDF files with ease!
aziz_muhd33@yahoo.co.in
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
55. Nuclear Shell Model
• Potential between nucleons can be studied by studying
bound states (pn, ppn, pnn, ppnn) or by scattering cross
sections: np -> np pp -> pp nD -> nD pD -> pD
• If had potential could solve Schrod. Eq. Don’t know
precise form but can make general approximation
• 3d Finite Well with little r-dependence (except at edge of
well)
• Almost spherically symmetric (fusion can be modeled as
deformations but we’ll skip)
• N-N interactions are limited (at high A) due to Pauli
exclusion. p + n -> p’ + n’ only if state is available
pdfMachine
A pdf writer that produces quality PDF files with ease!
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
56. Infinite Radial Well
2 d 2u 2 l ( l 1)
• Radial part of Scrod Eq
V ( r ) u Eu
2m dr 2 2m r2
u( r ) rR ( r ) P ( r ) 4u 2
• Easy to solve if l=0
2 n p2 ( k ) 2 ( hn) 2
u sin kr k E
2a 2m 2m 8ma 2
• For L>0, angular momentum term goes to infinity at r=0.
Reduces effective wavelength, giving higher energy
• Go to finite well. Wave function extends a bit outside well giving
longer effective wavelength and lower energy (ala 1D square
wells)
pdfMachine
• In nuceli, potential goes to infinity at r=0 (even with L=0) as that
A pdf writer that produces quality PDF files with ease!
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
would be equivalent to nucleon “inside” other nucleon
“Broadgun pdfMachine printer” and that’s it! Get yours now!
57. Angular part
• If V(r) then can separate variables y(r,q,f) = R(r)Y( q,f) have spherical
harmonics for angular wave function
• Angular momentum then quantized like in Hydrogen (except that L>0
for n=1, etc)
L2 l (l 1) 2 LZ m
l 0,1,2 m l l n r quantum#
• Energy doesn’t depend on m
• Energy increases with increasing n (same l)
• Energy increases with increasing l (same n)
• If both n,l vary then use experimental observation to determine
lower energy
• Energy will also depend on strong magnetic coupling between
pdfMachine
nucleons
A pdf writer that produces quality PDF files with ease!
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
• Fill up states separately for p,n
“Broadgun pdfMachine printer” and that’s it! Get yours now!
58. L,S,J Coupling: Atoms vs Nuclei
• ATOMS: If 2 or more electrons, Hund’s rules:
• Maximise total S for lowest E (S=1 if two)
• Maximise total L for lowest E (L=2 if 2 P)
• Energy split by total J (J=3,2,1 for S=1,L=2)
• NUCLEI: large self-coupling. Plus if 2 p (or 2 n) then will anti-align
giving a state with J=0, S=0, L=0
leftover “odd” p (or n) will have two possible
J = L + ½ or J = L – ½
higher J has lower energy
if there are both an odd P and an odd n (which is very rare in
stable) then add up Jn + Jp
• Atom called LS coupling nuclei called jj
pdfMachine
A pdf writer that produces quality PDF files with ease!
• Note that magnetic moments add differently as different g-factor
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
for p,n
59. Spin Coupling in Nuclei
• All nucleons in valence shell have same J
• Strong pairing causes Jz antiparallel (3 and -3) spin
wavefunction = antisymmetric space wavefunction =
symmetric
• This causes the N-N to be closer together and increases the
attractive force between them
• e-e in atoms opposite as repulsive force
• Can also see in scattering of polarized particles
• Even N, even Z nuclei. Total J=S=L=0 as all n,p paired off
• Even N, odd Z or odd N, even Z. nuclear spin and parity
determined by unpaired nucleon
• Odd N, odd Z. add together unpaired n,p
pdfMachine
• Explains ad hoc pairing term in mass formula
A pdf writer that produces quality PDF files with ease!
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
60. Energy Levels in Nuclei
• Levels in ascending order (both p,n)
State n L degeneracy(2j+1) sum
1S1/2 1 0 2 2***
1P3/2 1 1 4 6
1P1/2 1 1 2 8***
1D5/2 1 2 6 14
2S1/2 2 0 2 16
1D3/2 1 2 4 20***
1F7/2 1 3 8 28***
2P3/2 2 1 4 32
1F5/2 1 3 6 38
2P1/2 2 1 2 40
1G9/2 1 4 10 50***
*** “magic” number is where there is a large energy gap between a
pdfMachine
filled shell and the next level. More tightly bound nuclei. (all filled
A pdf writer that produces quality PDF files with ease!
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
subshells are slightly “magic”)
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
61. Magic Numbers
• Large energy gaps between some filled shells and next (unfilled)
shell give larger dE/A and more made during nucleosnthesis in
stars
# protons #neutrons
2 He 2 He-4
6 C 6 C-12
8 O 8 O-16
20 Ca 20
28 Ni 28 Cr-52(24,28)
50 Sn 50 Ni-78
82 Pb 82
126
136
• Ni-78 (2005) doubly magic. While it is unstable, it is the much
neutron rich.
pdfMachine
A pdf writer that produces quality PDF files with ease!
• Usually more isotopes if p or n are magic. Sn has 20 isotopes, 10
Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across
nearly all Windows platforms, simply open the document you want to convert, click “print”, select the
“Broadgun pdfMachine printer” and that’s it! Get yours now!
of which are stable