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Atoms and Nuclei
PA 322
Lecture 12
Unit 3:
Introduction to
nuclear properties
(continued)
(Reminder: http://www.star.le.ac.uk/~nrt3/322)
Topics
•  Basic nuclear properties
–  full list:
•  mass, charge, mass & charge distribution, radius,
abundances of isotopes (if nuclide stable), decay modes,
half-lives (if unstable), reaction modes, spin magnetic dipole
moment, electric quadrupole moment …
–  covered in last lecture: size, distribution of mass and charge
–  covered in this lecture
•  mass and abundances
•  binding energies
•  stability
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Lecture 12
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Mass and abundance of nuclides
•  Key measurements are:
–  mass of nuclide
–  mass number of nuclide = A = N + Z
–  relative abundances of different isotopes [same element (same
Z), but different A]
–  NB: nuclide mass ≠ mass number
•  Atomic weights of elements from simple chemical techniques and
knowledge of Avogadro’s Number
–  but these reflect only average over abundances of different
isotopes of same element
–  accuracy limited
•  Atomic numbers from, e.g., X-ray spectroscopy (wavelengths of Kα
lines ∝ 1/Z2)
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detector
Nuclide masses can be determined with
very high precision using mass
spectrometer
E
ion
source
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velocity
selector
Lecture 12
momentum selector
4
Mass spectrometer
Determination of nuclide masses and abundances
• 
• 
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Ions of the material to be tested are
produced by discharge or electron
bombardment
Ions enter “velocity selector” with range of
velocities v
–  orthogonal electric and magnetic fields
E and B1
–  only ions with v = vsel emerge: where
forces due to electric and magnetic fields
exactly cancel (other ions blocked).
detector
E
ion
source
velocity
selector
momentum selector
in practice B1 and B2 can be the same magnetic field
Ions with single velocity vsel enter momentum selector
–  uniform magnetic field B2 separates ions by momentum and hence mass
(since single value of velocity): produces trajectories with different radii r
–  final position of ion in detector thus determines nuclide mass
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Mass spectrometer
Determination of nuclide masses and abundances
• 
Velocity selection
–  force from electric field
–  force from magnetic field
FE = q E
FM = q (v x B1)
–  for orthogonal fields, forces cancel for vsel = E/B1
• 
Momentum selection
–  motion perpendicular to B field:
mvsel = qB2 r
mv2/r = qvB
r = (mvsel )/(qB2)
–  nuclide mass is thus uniquely determined by r
–  mass number directly obtained from mass: nearest integer with mass
expressed in u (1/12 12C mass)
–  mass spectrometer separates isotopes with different nuclide mass
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Mass spectrometer
Determination of nuclide masses and abundances
36Kr
Atomic weight = 83.798
• 
• 
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• 
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Measurements made with respect to 12C
Absolute accuracy ~10-6
Relative accuracy ~10-8 by comparing materials with similar mass
effective atomic weight of element = ave. over all isotopes
Lecture 12
7
Ball-park estimate of energy of nucleons
•  Nucleons to be confined to the atomic nucleus, i.e. with r ≤ Rnucleus
•  Uncertainty principle:
p is nucleon momentum
ℏ ≲ Δp Δx thus implies ℏ / Rnucleus ≲ Δp
putting p ~ Δp and E = p2/2mp E = ½ mpv2 = ½ (mpv)2 /mp
this gives E ~ ℏ2/ (2mpR2)
for R ~ 2 fm get E ~ 5 MeV
•  Simple estimate of (kinetic) energy of nucleon is E ~ 5 MeV
•  Compare to energy of outer electrons in atom E ~ 1-2 eV
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Origin of binding energy
•  Binding energy due to fact that nucleons are in potential well, ie.
have –ve potential energy
–  energy needs to be supplied to remove nucleons
net potential due to nuclear force (attractive) and
Coulomb force (repulsive between protons)
nuclear force at short range > Coulomb force
V(r)
-B
r
–  But also accounts for kinetic energy.
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Binding energy
•  Nuclide masses < sum of parts (if they were equal determining
accurate masses would not be very interesting)
•  Mass deficit due to binding energy of nucleus (mass energy of deficit
is the binding energy)
•  Mass energy of nuclide with atomic number Z and mass number A:
(mA – Zme) is quantity measured
in mass spectrometer
but mA is quantity usually tabulated
can be ignored,
typically ~10-6 of mc2
mc2 = mA c2 – Z mec2 + Be
atomic mass
energy
mass energy
of electrons
electronic
binding
energy
A
Z
•  Binding energy of a nuclide X is difference between its mass
energy and its constituent nucleons, ie.
B = [ Z mp + N mn – { m(AX) – Zme } ] c2
m = m(AX)
A
nucleons
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mass energy of nuclide
Lecture 12
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Binding energy
• 
Binding energy can be rewritten in terms of hydrogen atom masses:
B = [ Z m(1H) + N mn – m(AX)] c2
• 
Binding energy concept sometimes expressed as mass defect Δ where
Δ = (mA – A) c2
•  mass defect is directly related to binding energy, but with offset
• 
Example
16O: mass m = 15.994915 u (from Table in Krane)
A
Binding energy B = [8 mp + 8 mn – {15.994915 – 8 me}]
B = [8.05821176 + 8.06932008 - 15.994915 + 0.00438864] u
B = 0.1370 u = 127.62 MeV
B/A = 7.98 MeV/nucleon
[Mass defect Δ = 15.994915 u – 16 = -4.737 MeV]
1u = 931.50 MeV
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Binding energy per nucleon as function of A
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Binding energy dependence with mass number
• 
Dependence of binding energy per nucleon with mass number A
–  strong increase at low A
–  more gentle decline at high A
–  approx constant for most A at value of B/A~ 8 MeV
–  maximum binding energy per nucleon at A ~ 60 (close to Fe)
• 
Implications
–  combining two nuclei with low A increases B and thus releases energy:
FUSION!
(~4 MeV per nucleon)
–  splitting nucleus with high A increases B and thus releases energy:
FISSION!
(~ 1 MeV per nucleon)
–  processes do not occur spontaneously (fortunately!), require activation
energy
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neutron rich
stable nuclides are shaded black,
unstable (radioactive) nuclides are shaded grey
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Stability of nuclides
•  Overall pattern:
–  stable nuclides
•  for small Z, N~Z and A~2Z
•  for large Z, N>Z
•  most stable nuclei have even Z
and even N
•  significant number with even-odd
and odd-even configurations
•  only a very few odd-odd nuclei
are stable (e.g.21H 147 N )
–  unstable nuclides are typically those
furthest from line of stability
•  exhibit radioactivity, transformed
into different nuclides
PA322
Lecture 12
A
N
Z
number
stable
even
even
odd
even
odd
166
8
odd
even
odd
odd
even
57
53
15
Note, axes
flipped compared
to previous plot.
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Stability of nuclides
• 
Reasons for stability/instability
–  must be related to properties of nuclear and Coulomb forces (and
balance between them)
–  can be explored explicitly in terms of contributions to total binding
energy (e.g. the semi-empirical mass model which predicts B(A) )
–  for small Z, N~Z and A~2Z
•  similar numbers of protons and neutrons must be preferred
configuration
–  for large Z, N>Z
•  Coulomb repulsion increases with Z; additional neutrons needed to
dilute repulsive force
–  most stable nuclei have even Z and even N
–  significant number with even-odd and odd-even configurations
–  only a very few odd-odd nuclei are stable
•  related to properties of nuclear force:
–  binding energies must be Beven-even > Bodd-even ~ Beven-odd > Bodd-odd
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