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These lecture notes cover Chapter 10 of Nuclear Physics for Nuclear Engineers, which discusses nuclear properties. The notes cover the discovery of the nucleus, characteristics of nucleons, proton and neutron charge distributions, the nucleon-nucleon force, and nomenclature related to isotopes. The notes were created by Alex F Bielajew in 2012 for the Nuclear Engineering and Radiological Sciences course at the University of Michigan.
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aka Elements of Nuclear Engineering and Radiological Sciences IINERS 312
Nuclear Physics for Nuclear Engineers
Supplement toLecture Notes for Chapter 10: Nuclear Properties
(Krane II: Chapters 1 & 3)
Note: The lecture number corresponds directly to the chapter number in the online book.
The section numbers, and equation numbers correspond directly to those in the online book.
c
©
Alex F Bielajew 2012, Nuclear Engineering and Radiological Sciences, The University of Michigan
π
” scattering
Nuclear Engineering and Radiological SciencesErnest Marsden (right)and undergraduate (!)PostDoc Hans Geiger (left)experiment conducted by
NERS 312: Lecture 10, Slide # 1:10.
a.k.a.
the “nucleons”.
and discovered by Rutherford in 1920Protons were proposed by William Prout in 1815
later discovered by James Chadwick in 1932 In 1920, Rutherford proposed existence of the neutron
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 3:10.
Name
mass
charge
lifetime
magnetic moment
(symbol)
(MeV/c
2
e
(s)
μ N = e ℏ /
m
p
neutron (
n
proton (
p
stable
p Charge Distribution:
all +ve charge peaks at about 0.45 fm, with an exponential die-off to
fm
n
has a +ve charge that peaks at 0.24 fm, followed by a -ve shell that peaks at 0.95 fm
dies off exponentially, to about 2 fm
Graphs on the next 2 pages
Properties Common to both nucleons
Structure
Composite (quarks)
Radius
fm
Statistics
Fermi-Dirac (fermions)
Family
Baryons (3 quarks)
Intrinsic Spin
2 1
Active forces
Strong, electromagnetic, weak, gravity
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 4:10.
NERS 312: Lecture 10, Slide # 6:10.
but of opposite sign.particles, much like an electrostatic dipole field is created by two nearby, equal charges,The nucleon force is a “derivative force”, generated by the underlying constituent
The
n - n , n - p
, and
p
p
nuclear
forces are all almost identical.
V There is a strong short-range repulsive force, of the form:
rep
nn
r
nnrep
r
exp (
r/r
rep
nn
where
r
rep
nn
25 fm
V There is a strong medium-range attractive force, of the form:
att
nn
r
nnatt
r
exp (
r/r
att
nn
where
r
att
nn
36 fm
nnatt
nnrep
V The combined force is:
nn
r
nnrep
r
exp (
r/r
rep
nn
nnatt
r
exp (
r/r
nnatt
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 7:10.
it as a “contact” force. (Ping-pong balls covered in VaselineThe nuclear force bounds nucleons tightly, it is short-ranged. We can almost think if
TM
Nucleons are fermions, hence
n
’s and
p
’s tend to avoid each other.
The nuclear part of the nucleon-nucleon force is central.
lm
l
Protons are subject to a repulsive Coulomb force (also central).
Since
n
’s and
p
’s have magnetic moments, they are both subject to magnet and motion
v
) forces,
i.e.
, spin-spin, and spin-orbit effects. These have enormous impact
because the “magnets” are in close proximity.
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 9:10.
Z A
N
Isotope notation
Chemical symbol,
e.g.
Ca, Pb
Atomic mass number (sum of
n
’s and
p
’s in the nucleus)
Atomic number (or, proton number), the number of
p
’s in the nucleus
Neutron number, the number of
n
’s in the nucleus
Examples
23
He
1
2040
Ca
20
82208
Pb
126
Variants
40
Ca, Calcium-40, Ca-
Note that, once X (which encodes
) and
are given, the rest of the information is
redundant, since
. The full form is usually used only for emphasis.
isotope
Same
, different
e.g.
40
Ca and
41
Ca
Mnemonic: From Greek
isos
(same)
topos
(place) (coined by F. Soddy 1913)
i.e.
same place in the periodic table
isotone
Different
, same
e.g.
13
C and
12
Mnemonic: isoto
e and isoto
e (coined by K. Guggenheimer 1934)
isobar
Different
, and
, but same
e.g.
12
C and
12
Mnemonic: From Greek
isos
(same)
baros
(weight)
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 10:10.
As a nucleus gains nucleons, Coulomb repulsion, a long-range force, is felt by
all
the
protons in the nucleus.protons in the nucleus, that is, each proton feels the repulsion of all of the other
At some point, Coulomb repulsion overwhelms the nuclear
binding potential, and the nucleus can not be stable.
Nuclei can offset this instability by increasing the number of
n
’s, compared to the
number of
p
’s, thereby pushing the
p
’s to greater average radii. This strategy eventually
fails for
The heaviest isotope with an equal number of
n
’s and
p
’s is calcium. The heaviest
stable isotope is lead, with
nucleus will be discussed in the next section.is very nearly true. Exceptions will be discussed later in the course.) The radius of theThe strength of the nuclear force suggests that all nuclei are spherical in shape. (This
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 12:10.
Bang things together and interpret!
A: Q: How?
Bombard the nucleus with
e
−
’s (electrons).
Measure their deflection, from the
p
’s in the nucleus.
A: Q: Can you get information about the radius of the nucleus?
Yes! And as it turns out, that’s the best way to do it!
But
, you should account for a few things ... a few very important things.
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 13:10.
It starts with consideration of a factor we’ll call the “scattering amplitude”,
, where
k
i
k
f
F
e
i
~ k
f
·
~x
~x
e
i
~ k
i
·
~x
where ...
e
i
~ k
i
·
~x
the initial unscattered
e
−
wavefunction
e
i
~ k
f
·
~x
the final scattered
e
−
wavefunction
k
i
the initial
e
−
wavenumber
k
f
the final
e
−
wavenumber
~x
in the nucleusthe electrostatic Coulomb potential arising from the +ve charge distribution
F
a proportionality constant to be determined later
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 15:10.
F Evaluating ...
k
i
k
f
F
d
~x e
−
i
k~
f
·
~x
~x
e
i
k~
i
·
~x
F
d
~x V
x
e
i
(
~ k
i
−
~ k
f
)
·
~x
F
d
~x V
x
e
i~
q
·
~x
where
~q
k
i
k
f
is called the
momentum transfer
It is the momentum taken (transferred) from
k
i
to produce
k
f
, since
k
f
k
i
q
We see that
is a function of the momentum transform alone:
~q
F
d
~x V
~x
e
i~
q
·
~x
Nuclear Engineering and Radiological Sciencespotential. We are now, hopefully, in familiar mathematical territory!We also see that scattering amplitude is proportional to the 3D Fourier Transform of the
NERS 312: Lecture 10, Slide # 16:10.
In this application...
~x
is treated as a classical, continuous charge distribution.
In fact, the operator
wavefunction.is made up of the quantum mechanical E&M operator over the composite proton
forces are relatively weak.First Born Approximation. It happens to work for this application because the E&M
Nuclear Engineering and Radiological SciencesReturning to our analysis ...
NERS 312: Lecture 10, Slide # 18:10.
V The scattering potential takes the form:
~x
Ze
2
πǫ
0
d
~x
′
ρ
p
x
′
x
~x
′
ρ
p
x
′
the “number” density of protons in the nucleus, normalized so that:
d
~x
′
ρ
p
x
′
The potential at
~x
arises from the electrostatic attraction of the elemental charges in
d
x
F Putting it all together:integrated over the nucleus.
~q
F
Ze
2
πǫ
0
d
x
d
x
′
ρ
p
~x
′
~x
x
′
e
i~
q
·
~ x
We choose the constant of proportionality in
~q
, to require that
The
motivation for this choice is that, when
~q
, the charge distribution is known to have no
F (10.5) aseffect on the projectile. If a potential has no effect on the projectile, then we can rewrite
F
Ze
2
πǫ
0
d
~x
d
~x
′
ρ
p
~x
′
~x
x
′
or
Nuclear Engineering and Radiological Sciences
NERS 312: Lecture 10, Slide # 19:10.
