Nuclear PhysicsAbout Units
Energy - electron-volt
1 electron-volt = kinetic energy of an electron when moving through potential difference of 1 Volt;
1 eV = 1.6 × 10-19 Joules
1 kW•hr = 3.6 × 106 Joules = 2.25 × 1025 eV
1 MeV = 106 eV, 1 GeV= 109 eV, 1 TeV = 1012 eV
mass - eV/c2
electron mass = 0.511 MeV/c2
neutron mass = 939.6 MeV/c2
1 eV/c = 5.3 × 10-28 kg m/s
momentum of baseball at 80 mi/hr 5.29 kgm/s 9.9 × 1027 eV/c
Distance
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A. Becquerel, Maria Curie, Pierre Curie(1896 – 1898):
also other heavy elements (thorium, radium) show radioactivity
three kinds of radiation, with different penetrating power (i.e. amount of material necessary to attenuate beam):
“Alpha (a) rays” (least penetrating – stopped by paper)
“Beta (b) rays” (need 2mm lead to absorb)
“Gamma (g) rays” (need several cm of lead to be attenuated)
three kinds of rays have different electrical charge: a: +, b: -, g: 0
Identification of radiation:
Ernest Rutherford (1899)
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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
measure scattering angles of particles with scintillating screen (ZnS)
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Geiger Marsden experiment: result
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
Rutherford atom: positive charge in nucleus
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bombard light elements (e.g. 49Be) with alpha particles neutral radiation emitted
Irène and Frederic Joliot-Curie (1931)
pass radiation released from Be target through paraffin wax protons with energies up to 5.7 MeV released
if neutral radiation = photons, their energy would have to be 50 MeV -- puzzle
puzzle solved by James Chadwick (1932):
“assume that radiation is not quantum radiation, but a neutral particle with mass approximately equal to that of the proton”
identified with the “neutron” suggested by Rutherford in 1920
observed reaction was: (24He++) + 49Be 613C*
613C* 612C + n
apparent “non-conservation” of energy
Wolfgang Pauli predicted a light, neutral, feebly interacting particle (called it neutron, later called neutrino by Fermi)
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Ea, g = Ei – Ef
But b spectrum continuous
Energy conservation violated??
Bohr:: “At the present stage of atomic theory, however, we may say that we have no argument, either empirical or theoretical, for upholding the energy principle in the case of β-ray disintegrations”
F. A. Scott, Phys. Rev. 48, 391 (1935)
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Positron track going
upward through lead
one of the greatest physicists of the 20th century
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R = r0 A1/3, r0 = 1.2 x 10-15 m = 1.2 fm;
i.e. ≈ 0.15 nucleons / fm3
generally spherical shape, almost uniform density;
made up of protons and neutrons
protons and neutron -- “nucleons”; are fermions (spin ½), have magnetic moment
nucleons confined to small region (“potential well”)
occupy discrete energy levels
two distinct (but similar) sets of energy levels, one for protons, one for neutrons
proton energy levels slightly higher than those of neutrons (electrostatic repulsion)
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Find the ratio of the radii for the following nuclei:
1H, 12C, 56Fe, 208Pb, 238U
1 : 2.89 : 3.83 : 5.92 : 6.20
ro = 1.2 x 10-15 m
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A range from 1 ((hydrogen) to 238 (Uranium)
N = neutron number = A-Z
nomenclature:
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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 126C, with A = 12, Z = 6:
1 amu = (mass of 126C atom)/12
1 amu = 1.66 x 10-27kg
1 amu = 931.494 MeV/c2
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 e/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 e/2mn
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energy units MeV/c2
all nuclei of certain element contain same number of protons, but may contain different number of neutrons
examples:
deuterium, heavy hydrogen 2D or 2H; heavy water = D2O (0.015% of natural water)
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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
(cp) = hc/(2 x) = hc/(2 r)
(cp) = 6.63x10-34 Js * 3x108 m/s / (2 * 1.33x10-15 m)
(cp) = 2.38x10-11 J = 148.6 MeV
E = p2/2m = (cp)2/2mc2 = (148.6 MeV)2/(2*940 MeV) = 11.7 MeV
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Mass of Nucleus Z(mp) + N(mn)
“mass defect” m = difference between mass of nucleus and mass of constituents
energy defect = binding energy EB EB = m c2
binding energy = amount of energy that must be invested to break up nucleus into its constituents
binding energy per nucleon = EB /A
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Nuclear Binding Energy
The difference between the energy (or mass) of the nucleus and the energy (or mass) of its constituent neutrons and protons.
= the energy needed to break the nucleus apart.
Average binding energy per nucleon = total binding energy divided by the number of nucleons (A).
Example: Fe-56
42He 4.00153 amu
168O 15.991 amu
5626Fe 55.922 amu
23592U 234.995 amu
note:
Gamma Decay
AZ* AZ +
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AZ A(Z+1) + e- + an anti-neutrino
A neutron has converted into a proton, electron and an anti-neutrino.
Positron Decay
AZ A(Z-1) + e+ + a neutrino
A proton has converted into a neutron, positron and a neutrino.
Electron Capture
AZ + e- A(Z-1) + a neutrino
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Rate of decay number N of nuclei
Solution of diff. equation (N0 = nb. of nuclei at t=0)
Mean life = 1/
atomic nuclei small -- about 1 to 8fm
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
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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;
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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 nucl. 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
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liquid drop model (Bohr, Bethe, Weizsäcker):
nucleus = drop of incompressible nuclear fluid.
fluid made of nucleons, nucleons interact strongly (by nuclear force) with each other, just like molecules in a drop of liquid.
introduced to explain binding energy and mass of nuclei
predicts generally spherical shape of nuclei
good qualitative description of fission of large nuclei
provides good empirical description of binding energy vs A
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Bethe - Weizsäcker formula:
an empirically refined form of the liquid drop model for the binding energy of a nucleus of mass number A with Z protons and N neutrons
binding energy has five terms describing different aspects of the binding of all the nucleons:
volume energy
surface energy
an asymmetry term (N vs Z)
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assume nucleons move inside nucleus without interacting with each other
Fermi- gas model:
Protons and neutrons move freely within nuclear volume, considered a rectangular box
Protons and neutrons are distinguishable and so move in separate potential wells
Shell Model
formulated (independently) by Hans Jensen and Maria Goeppert-Mayer
Each nucleon (proton or neutron) moves in the average potential of remaining nucleons, assumed to be spherically symmetric.
Also takes account of the interaction between a nucleon’s spin and its angular momentum (“spin-orbit coupling”)
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In each potential well, the lowest energy states are occupied.
Because of the Coulomb repulsion the proton well is shallower than that of the neutron.
But the nuclear energy is minimized when the maximum energy level is about the same for protons and neutrons
Therefore, as Z increases we would expect nuclei to contain progressively more neutrons than protons.
U has A = 238, Z = 92
Potential well
collective model is “eclectic”, combining aspects of other models
consider nucleus as composed of “stable core” of closed shells, plus additional nucleons outside of core
additional nucleons move in potential well due to interaction with the core
interaction of external nucleons with the core agitate core – set up rotational and vibrational motions in core, similar to those that occur in droplets
gives best quantitative description of nuclei
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energy released if break up into two medium sized nuclei
“fission”
light nuclei:
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Sun’s power output
3.826 x 1026 Watts
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e+ + e- → g + g
2H + 1H → 3He + g
Deuterium creation
3He creation
4He creation
4H → 4He
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Hydrogen fusion
Sir Fred Hoyle
Summary
nuclei made of protons and neutrons, held together by short-range strong nuclear force
models describe most observed features, still being tweaked and modified to incorporate newest observations
no full-fledged theory of nucleons yet
development of nuclear theory based on QCD has begun
nuclear fusion is the process of energy production of Sun and other stars
we (solar system with all that’s in it) are made of debris from dying stars
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