Transcript Nuclear Physics

Nuclear Physics
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Topics
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Composition of the Nucleus
Ground State Properties
Nuclear Structure
Binding energy
Nuclear Models
Summary
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Geiger, Marsden, Rutherford expt.
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(Geiger, Marsden, 1906 - 1911) (interpreted by Rutherford, 1911)
get  particles from radioactive source
make “beam” of particles using “collimators”
(lead plates with holes in them, holes aligned in straight line)
bombard foils of gold, silver, copper with beam
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measure scattering angles of particles with scintillating screen (ZnS)
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Geiger Marsden experiment: result
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most particles only slightly deflected (i.e. by small angles),
but some by large angles - even backward
measured angular distribution of scattered particles did not
agree with expectations from Thomson model (only small
angles expected),
but did agree with that expected from scattering on small,
dense positively charged nucleus with diameter
< 10-14
m, surrounded by electrons at 10-10 m
Ernest
Rutherford
1871-1937
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Composition of the Nucleus -- 1
From the experiments of Geiger and Marsden in
1911, it became clear that most of the mass of
an atom is contained within a nucleus of size
~ 1 fm (=10-15 m).
In 1932, the neutron was discovered by Chadwick,
the positron by Anderson and the first nuclear
reaction with protons was observed by Cockcroft
and Walton.
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Composition of the Nucleus -- 2
A nucleus is a quantum system consisting of
neutrons and protons, held together by a strong
nuclear force.
A nucleus of a given species, called a nuclide, is
defined by its atomic number Z, that is, by the
number of protons it contains.
N is the number of neutrons in the nucleus and
A = N + Z is the mass number. For example, 15O
has A = 15, N = 7 and Z = 8.
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Composition of the Nucleus -- 3
Neutrons and protons are referred to collectively
as nucleons.
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Ground State Properties -- 1
Nuclei with the same atomic number, but which
differ in mass number, e.g., 15O and 16O are
called isotopes.
examples:
deuterium, heavy hydrogen 2D or 2H;
heavy water = D2O (0.015% of natural water)
U- 235 (0.7% of natural U), U-238 (99.3% of natural U),
If nuclei have the same neutron number N, e.g.,
13C and 14N they are called isotones.
Those with the same mass number, e.g., 14C and
14N are called isobars.
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Ground State Properties -- 2
Nuclear Sizes – The size of nuclei can be inferred
in many ways. One way is to use mirror nuclides:
those with Z and N numbers switched, for
example:
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15
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O 
8p + 7n
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N
7p + 8n
If we assume that the nuclear force is the same
for protons and neutrons, then the energy of
these nuclei will differ by their electrostatic energy.
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Ground State Properties - 3
Let’s model the positive charge q of a nucleus as
a ball of uniform charge of radius R. The potential
energy of this ball of charge is given by
2
3 q
U k
5 R
Extra Credit: Prove this. Hint: consider the
potential energy between a sphere of charge
of radius r and a thin shell of charge of radius
r and thickness dr, then integrate. Due Nov. 9
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Ground State Properties - 4
The nucleus 15O is radioactive and decays as
follows 15O  15N + e+ + . The energy
difference between the nuclei is
2
3 e
U  k  Z 2  ( Z  1)2 
5 R
From numerous measurements of this energy
difference it has been deduced that
R  R0 A
1/ 3
with R0 = 1.2 ± 0.2 fm
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Ground State Properties - 5
Another way to measure
nuclear radii is to scatter
electrons off nuclei and
measure the resulting
diffraction pattern of
the scattered electrons.
The first minimum of
this pattern occurs at
 hc  1
sin   0.16  
 pc  R
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Ground State Properties - 6
The electron scattering
experiments were first
carried about by
Robert Hofstadter in
the 1950s at SLAC.
These experiments gave
information about the
charge profile of nuclei,
as shown in the figure.
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Ground State Properties - 7
The results of Hofstadter’s experiments
showed that the charge distribution of a
nucleus can be described as a ball of uniform
charge density, which, near the surface,
decreases to zero over a zone of thickness
t = 2.4 ± 0.3 fm.
The radius measurements obtained by his team
were consistent with those deduced from the
mirror nuclei.
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Ground State Properties - 8
Nuclear Density – Since the radius of a nucleus is
proportional to A1/3, the density of nuclear
matter is roughly independent of the size of the
nucleus. Consequently, nuclear matter behaves
rather like a liquid with the enormously high
density of 1017 kg/m3.
A mere 1 cubic millimeter of nuclear matter would
weigh as much as a full supertanker!
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Ground State Properties - 9
Nuclear Energies – The electrostatic energy can
be written as
2
2
3 e 2 3 ke c 2
U k Z 
Z
5 R
5 c R
3 c 2

Z
5 R
where α ~ 1/137 and ħc = 197 MeV.fm
For 16O, Z = 8, R = 1.2A1/3 = 3 fm, therefore,
U = 18.3 MeV
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Ground State Properties - 10
Nuclear Pressures – The pressure can be computed
using
U U R 1 U
P
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V R V 3 V
For 16O, U = 18.3 MeV, R = 3 fm, so V = 116 fm3.
Therefore,
P
= (1/3) x (U/V)
= (1/3) x (18.3 MeV/116 fm3)
= 0.053 MeV/fm3
= 8.4 x 1030 Pa
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Nuclear Structure - 1
The neutron
number, N,
increases
faster
than the
atomic
number, Z.
Why? The
exclusion
principle
Line of
stability
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Nuclear Structure - 2
A system with 7 neutrons has a higher overall
energy than one with 4 neutrons and 3 protons
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Nuclear Structure - 3
Moreover, for large Z, because neutrons are electrically
neutral, less energy is needed to add 2 neutrons than to add
1 neutron and
1 proton
Therefore,
N-Z
increases
with Z
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Atomic mass unit
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“atomic number” Z
 Number of protons in nucleus
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Mass Number A
 Number of protons and neutrons in nucleus
 Atomic mass unit is defined in terms of the
mass of the atom 126C (A = 12, Z = 6):
 1 amu = (mass of 126C atom)/12
 1 amu = 1.66 x 10-27 kg
 1 amu = 931.494 MeV/c2
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Properties of Nucleons
 Proton
Charge = 1 elementary charge e = 1.602 x 10-19 C
 Mass = 1.673 x 10-27 kg = 938.27 MeV/c2
= 1.007825 u = 1836 me
 spin ½, magnetic moment 2.79 eħ/2mp
Neutron
 Charge = 0
 Mass = 1.675 x 10-27 kg = 939.6 MeV/c2
= 1.008665 u = 1839 me
 spin ½, magnetic moment -1.9 eħ/2mn
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Nuclear energy levels: example
Problem: Estimate the lowest possible energy of a neutron
contained in a typical nucleus of radius 1.33×10-15 m.
E = p2/2m = (cp)2/2mc2
x p = h/2

x (cp) = hc/2 = ħc
(cp) = hc/(2 x) = hc/(2 r)
(cp) = 4.1357x10-15 eVs * 3x108 m/s / (2 * 1.33x10-15 m)
(cp) = 1.485x108 eV = 148.5 MeV
E = p2/2m = (cp)2/2mc2 = (148.5 MeV)2/(2*940 MeV) = 11.7 MeV
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Nuclear Masses, binding energy
 Mass of Nucleus  Z(mp) + N(mn)
Mass defect m = difference between mass of
atom and mass of constituents
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 Binding energy EB = energy defect = m c =
amount of energy that must be invested to break
up atom into its constituents
Example: mass(7P + 7N + 7e) – mass(147N)
= 7(1.00728 + 1.00866 + 0.00055)
– 14.003074 = 0.1124 u
 binding energy per nucleon = EB /A
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Binding Energies
Iron is the most tightly bound
nucleus. This fact is very
important in stellar evolution.
B  Zmpc  Nmnc  M Ac
2
2
2
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Nuclear (“strong”) force - 1
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atomic nuclei small -- about 1 to 8 fm
at small distance, electrostatic repulsive forces
are of macroscopic size (10 – 100 N)
there must be short-range attractive force
between nucleons -- the “strong force”
strong force essentially charge-independent
 “mirror nuclei” have almost identical binding energies
 mirror nuclei = nuclei for which n  p or p  n
(e.g. 3He and 3H, 7Be and 7Li, 35Cl and 35Ar);
slight differences due to electrostatic repulsion
strong force must have very short range – << atomic size,
otherwise isotopes would not have same chemical
properties
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Strong force -- 2
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range: fades away at distance ≈ 3fm
 force between 2 nucleons at 2fm distance ≈ 2000N
 nucleons on one side of U nucleus hardly affected by
nucleons on other side
experimental evidence for nuclear force from scattering
experiments;
 low energy p or  scattering: scattered particles
unaffected by nuclear force
 high energy p or  scattering: particles can overcome
electrostatic repulsion and can penetrate deep enough
to enter range of nuclear force
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N-Z and binding energy vs A
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small nuclei (A<10):
 All nucleons are within range of strong force
exerted by all other nucleons;
 add another nucleon  enhance overall cohesive force
 EB rises sharply with increase in A
medium size nuclei (10 < A < 60)
 nucleons on one side are at edge of nuclear force range from nucleons
on other side  each add’l nucleon gives diminishing return in terms
of binding energy  slow rise of EB /A
heavy nuclei (A>60)
 adding more nucleons does not increase overall cohesion due to
nuclear attraction
 Repulsive electrostatic forces (infinite range!) begin to have stronger
effect
 N-Z must be bigger for heavy nuclei (neutrons provide attraction
without electrostatic repulsion
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 heaviest stable nucleus:
Bi
– all nuclei heavier than 209Bi are unstable
(radioactive)
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Nuclear Models
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Liquid Drop Model (Bohr, Bethe, Weizsäcker):
 Nucleus described as a drop of incompressible
nuclear fluid interacting via the strong force
 Predicts spherical shape of nuclei
 Qualitative description of fission of large nuclei
 Good description of binding energy vs A
Fermi Gas Model
 Neutrons and protons described as a free gas
Shell Model (Hans Jensen, Maria Goeppert-Mayer)
 Similar to shell description of atoms
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Summary
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Nuclei are made of protons and nucleons and have
radii that scale roughly as A1/3, where A is the mass
number.
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The density of nuclear matter, 1017 kg/m3, is roughly
independent of the size of the nucleus
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The nuclear energy scale is of order 10 MeV
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High Z nuclei tend to be neutron-rich
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