binding energy and nuclear forceswimu/educ/qb_lecture_12-2014.pdfthe higher the binding energy per...
TRANSCRIPT
The force that binds the nucleons together is
called the strong nuclear force. It is a very
strong, but short-range, force. It is essentially
zero if the nucleons are more than about 10-15
m apart. The Coulomb force is long-range; this
is why extra neutrons are needed for stability
in high-Z nuclei.
Binding Energy and Nuclear Forces
Calculate that at 1 fm: MeV44.14
e
0
2
r
VCoulomb
Nuclei: protons and neutrons.
Proton has positive charge;
mn = 1.67262 x 10-27 kg
Neutron is electrically neutral;
mn = 1.67493 x 10-27 kg
Structure and Properties of the Nucleus
Number of protons: atomic number, Z
Number of nucleons: atomic mass number, A
Neutron number: N = A - Z
Symbol:
Isotopes
(85) 61 672 1836.152e
p
m
m
No theory
Because of wave–particle duality, the size of the nucleus is somewhat
fuzzy. Measurements of high-energy electron scattering yield:
Extreme density
Masses scale: carbon-12 atom, 12 u
u is a unified atomic mass unit.
Magnetic moments
Protons and neutrons are not point particles
L
g BL
For a classical point particle
For a relativistic spin (point)
S
g BSS 2Sg
1Lg
Define nuclear magnetic moments
(unit = the nuclear magneton)
pN
M
e
2
Proton magnetic moment
Np 7928.2
Neutron magnetic moment
Nn 9135.1
Binding Energy and Einsteins E=mc2
2atomnucleusatom cMZmME eb
Binding energy affects mass of composite particle
mH-atom < mproton + melectron
mH-atom(n=1 state) < mH-atom(n=2 state)
These effects are much larger in the realm of nuclei
than in that of electromagnetism
2nucleusnucleus cMNMZME npb
(mind the signs !)
The total mass of a stable nucleus is always less than the sum of the
masses of its separate protons and neutrons.
Binding Energy and Nuclear Forces
2nucleusnucleus cMNMZME npb
2atomnucleus cMZmNMZME enpb
Here we neglect of the binding energy inside the atom (eV).
2A1A XHX cMNMZME nb
Mass of neutral atomBinding of nucleus
This can be used to calculate Eb/A: binding energy per nucleus
From observation: binding energy per nucleon: Eb/A
Binding Energy and Nuclear Forces
Note:
a is point of
relative stability
From observation:
The higher the binding
energy per nucleon, the
more stable the nucleus.
More massive nuclei
require extra neutrons to
overcome the Coulomb
repulsion of the protons
in order to be stable.
Binding Energy and Nuclear Forces
The semi-empirical mass formula
Von Weizsäcker
Liquid drop model
5
2
43/1
2
33/2
21A 2/
X
A
ZAa
A
ZaAaAaEb
Volume term
AR ~~ 3Surface correction:
on the outer surface (4R2)
the binding is less, because
There are no particles to
contribute
3/22 ~~ AR
Addition
Mev39
Mev80.94
Mev7105.0
Mev81.17
Mev76.15
5
4
3
2
1
a
a
a
a
a
Good fit for:
Note:
3/1~ AR
The semi-empirical mass formula
Von Weizsäcker
Liquid drop model
5
2
43/1
2
33/2
21A 2/
X
A
ZAa
A
ZaAaAaEb
Coulomb energy stored
in a uniform solid sphere
of charge Ze and radius R
(negative binding)3/1
2
~A
Z
R
ZeVrep
0
2
45
3
4/35
A
a ee
oo
0eooe
“pairing energy”
(not further discussed)
Addition
Protons and Neutrons in the Fermi-gas model
A
ZAa
2
4
2/
Addition
Both protons and neutrons cannot
fill the lowest “orbitals”
because the have to follow the
Pauli exclusion principle
“ Symmetry
Energy”
preference for
NZ
Based on this concept:
the “nuclear shell model”
"for the discovery concerning
nuclear shell structure"
Maria Goeppert-Mayer
Isobars and the mass equation
Addition
25
2
43/1
2
33/2
211A /
2/)(HX c
A
ZAa
A
ZaAaAaMZAZMM n
Mass of nucleus:
Isobar:
A
AZa
A
ZacMMc
Z
Mn
2/2H 43/13
212
(even-odd
separate
due to 5 term)
Minimum charge Z:
3/2
34
214 H
2 Aaa
cMMaAZ n
A
Valley of stability
Unstable elements
97Tc t = 2 x 106 y99Tc t = 2 x 105 y
145Pm t = 18 y146Pm t = 5.5 y
Stable/Meta-stable elements126Te stable, 18.95 %
128Te t > 5 x 1024 y
130Te t = 2.5 x 1021 y
235U t = 7 x 108 y238U t = 4.5 x 109 y
233U t = 1.6 x 105 y
Why do nuclei decay radioactivity