PHYS 3446 – Lecture #6
Monday, Feb. 7, 2005
Dr. Jae Yu
1. Nature of the Nuclear Force
• Short Range Nature of the Nuclear Force
• Shape of the Nuclear Potential
• Yukawa Potential
• Range of Yukawa Potential
2. Nuclear Models
• Liquid Drop Model
• Fermi-gas Model
• Shell Model
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
1
Announcements
• How many of you did send an account request to Patrick at
(mcquigan@cse.uta.edu)?
– I was told that only 7 of you have contacted him for accounts.
– There will be a linux and root tutorial session next Wednesday,
Feb. 16, for your class projects.
– You must make the request for the account by this Wednesday.
• First term exam
– Date and time: 1:00 – 2:30pm, Monday, Feb. 21
– Location: SH125
– Covers: Appendix A + from CH1 to what we cover next Monday,
Feb. 14
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
2
Nuclear Properties: Binding Energy
• The mass deficit
M  A, Z   M  A, Z   Zm p   A  Z  mn
• Is always negative and is proportional to the nuclear
binding energy
• How are the BE and mass deficit related?
B.E  M  A, Z  c
2
• What is the physical meaning of BE?
– A minimum energy required to release all nucleons from a
nucleus
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
3
Nuclear Properties: Binding Energy
• de Broglie’s wavelength:
– Where is the Planck’s constant
– And is the reduced wave length

p
• Assuming 8MeV was given to a nucleon (m~940MeV),
the wavelength is

p

197 Mev  fm

 1.6 fm

2  940  8
2mT
2mc 2T
c
• Makes sense for nucleons to be inside a nucleus since
the size is small.
• If it were electron with 8MeV, the wavelength is ~10fm,
a whole lot larger than a nucleus.
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
4
Nuclear Properties: Sizes
• Another way is to use the strong nuclear force using sufficiently
energetic strongly interacting particles (p mesons or protons, etc)
– What is the advantage of using these particles?
• If energy is high, Coulomb interaction can be neglected
• These particles readily interact with nuclei, getting “absorbed” into the nucleus
– These interactions can be treated the same way as the light absorptions resulting
in diffraction, similar to that of light passing through gratings or slits
• The size of a nucleus can be inferred from the diffraction pattern
• From all these phenomenological investigation provided the simple
formula for the radius of the nucleus to its number of nucleons or
atomic number, A:
R  r0 A  1.2  10
13
13
A cm  1.2 A fm
13
13
How would you interpret this formula?
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
5
Nuclear Properties: Magnetic Dipole Moments
• Every charged particle has a magnetic dipole
moment associated with its spin
e
g
2mc
S
• e, m and S are the charge, mass and the intrinsic
spin of the charged particle
• Constant g is called Lande factor with its value:
–
–
g  2: for a point like particle, such as the electron
g  2: Particle
possesses an anomalous magnetic
moment, an indication of having a substructure
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
6
Nuclear Properties: Magnetic Dipole Moments
• For electrons, e~B, where B is Bohr Magneton
B 
e
 5.79 1011 MeV/T
2me c
• For nucleons, magnetic dipole moment is measured in nuclear
e
magneton, defined using proton mass
N 
2m p c
• Magnetic moment of proton and neutron are:
n  1.91 N
 p  2.79 N
• What important information do you get from these?
– The Lande factors of the nucleons deviate significantly from 2.
• Strong indication of substructure
– An electrically neutral neutron has a significant magnetic moment
• Must have extended charge distributions
• Measurements show that mangetic moment of nuclei lie -3N~10N
– Indication of strong pairing
– Electrons cannot reside in nucleus
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
7
Nature of the Nuclear Force
• Scattering experiments helped to
– Determine the properties of nuclei
– More global information on the characteristics of the
nuclear force
• From what we have learned, it is clear that there is no
classical analog to nuclear force
– Gravitational force is too weak to provide the binding
– Can’t have an electromagnetic origin
• Deuteron nucleus has one neutron and one proton
• Coulomb force destabilizes the nucleus
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
8
Short-range Nature of the Nuclear Force
• Atomic structure is well explained by the electromagnetic
interaction
– Thus the range of nucleus cannot be much greater than the
radius of the nucleus
– Nuclear force range ~ 10-13 – 10-12cm
• Binding energy is constant per each nucleon, essentially
independent of the size of the nucleus
– If the nuclear force is long-ranged, like the Coulomb force
– For A nucleons, there would be ½ A(A-1) pair-wise interactions
– Thus, the BE which reflects all possible interactions among the
nucleons would grow as a function of A
B  A( A  1)
Monday, Feb. 7, 2005
For large A
PHYS 3446, Spring 2005
Jae Yu
B
A
A
9
Short-range Nature of the Nuclear Force
• If the nuclear force is long-ranged and is independent of the
presence of other nucleons, BE per nucleon will increase
linearly with A
– This is because long-range forces do not saturate
– Since any single particle can interact with as many other particle
as are available
Binding becomes tighter as the number of interacting objects
increases
– The size of the interacting region stays fairly constant
– Atoms with large number of electrons have the sizes compatible to
those with smaller number of electrons
• Long-rangeness of nuclear force is disputed by the
experimental measurement that the BE/nucleon stays
constant
– Nuclear force must saturate
– Any given nucleon can only interact with a finite number of
Monday,nucleons
Feb. 7, 2005 in its vicinity PHYS 3446, Spring 2005
Jae Yu
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Short-range Nature of the Nuclear Force
• What does adding more nucleons to a nucleus do?
– Only increases the size of the nucleus
• Recall that R ~ A1/3
– The size of a nucleus grows slowly with A and keep the
nuclear density constant
Another supporting evidence of short-range nature of
nuclear force
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
11
Shape of the Nuclear Potential
• Nuclear force keeps the nucleons within the nucleus.
– What does this tell you about the nature of the nuclear force?
It must be attractive!!
• However, scattering Experiments with high energy
revealed a repulsive core!!
– Below a certain length scale, the nuclear force changes from
attractive to repulsive.
– What does this tell you?
• Nucleons have a substructure….
• This feature is good, why?
– If the nuclear force were attractive at all distances, the nucleus
would collapse in on itself.
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
12
Shape of the Nuclear Potential
• We can turn these
behaviors into a squarewell potential
– For low energy particles,
the repulsive core can be
ignored, why?
• This model is too
simplistic, since there
are too many abrupt
changes in potential.
– There would be
additional effects by the
Coulomb force
Monday, Feb. 7, 2005
Attractive
force
Repulsive
Core
PHYS 3446, Spring 2005
Jae Yu
13
Nuclear Potential w/ Coulomb Corrections
Results in
• Classically an incident proton with total energy E0 cannot be closer
than r=r0. Why?
– For R<r<r0, V(r) >E0 and KE<0  Physically impossible
• What about a neutron?
– Could penetrate into the nuclear center.
• Low energy scattering experiment did not provide the exact shape of
the potential but the range and height of the potential
• The square-well shape provides a good phenomenological description
of the nuclear force.
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
14
Nuclear Potential
• Description of nuclear potential using a square well shape
suggests the basis of quantum theory with discrete energy
levels and corresponding bound state as in atoms
– Presence of such nuclear quantum states have been confirmed
through
• Scattering experiments
• Studies of the energies emitted in nuclear radiation
• Studies of mirror nuclei and the scattering of protons and
neutrons demonstrate
– Once Coulomb effects have been corrected, the forces between
two neutrons, two protons and a proton and a neutron are the
same  Nuclear force is charge independent!!
– Inferred as charge independence of nuclear force.
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
15
Nuclear Potential
• Strong nuclear force is independent of the electric
charge carried by nucleons
– Concept of strong isotopic-spin symmetry.
– Under this symmetry, proton and neutron are the two
different iso-spin state of the same particle, nucleon
– If Coulomb effect can be turned off, protons and
neutrons would be indistinguishable in their nuclear
interactions
– This is analogues to the indistinguishability of spin up
and down states in the absence of a magnetic field!!
• Iso-spin symmetry!!!
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
16
Range of Nuclear Force
• EM force can be understood as a result of a photon
exchange
– Photon propagation is described by the Maxwell’s equation
– Photons propagate at the speed of light.
– What does this tell you about the mass of the photon?
• Massless
• Coulomb potential is expressed as
1
V r  
r
• What does this tell you about the range of the
Coulomb force?
– Long range. Why?
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
17
Yukawa Potential
• For massive particle exchanges, the potential takes
the form
mc
 r
V r  
e
r
– What is the mass, m, in this expression?
• Mass of the particle exchanged in the interaction, or the force
mediator
• This form of potential is called Yukawa Potential
– Formulated by Hideki Yukawa in 1934
• In the limit m 0, Yukawa potential turns into
Coulomb
potential
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
18
Ranges in Yukawa Potential
• From the form of the Yukawa potential
V r  
e

mc
r
r
er

r
• The range of the interaction is given by some characteristic

value of r, Compton wavelength of the mediator with mass, m:
mc
• Thus once the mass of the mediator is known, range can be
predicted or vise versa
• For nuclear force, range is about 1.2x10-13cm, thus the mass of
the mediator becomes:
c 197 MeV  fm
2
mc 

 164 MeV
1.2 fm
• This is close to the mass of a well known p meson (pion)
m
p
m
p
2
 139.6 MeV / c ; m
p0
 135MeV / c
2
• Thus, it was thought that p are the mediators of the nuclear
force
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
19
Assignments
1. End of the chapter problems:
•
2.2, 2.5, 2.9.
2. Draw Yukawa potential for particles with the
following masses as a function of the radial
distance, r, in the range of 10-14 – 10-20 m in a semilogarithmic scale.
•
•
•
•
130 MeV/c2
80 GeV/c2
115 GeV/c2
Due for these homework problems is next Monday,
Feb. 16.
Monday, Feb. 7, 2005
PHYS 3446, Spring 2005
Jae Yu
20