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Nuclear Physics Topics

- Composition of Nucleus
- features of nuclei
- Nuclear Models
- nuclear energy
- Fission
- Fusion
- Summary

About Units

- Energy - electron-volt
- 1 electron-volt = kinetic energy of an electron when moving through potential difference of 1 Volt;
- 1 eV = 1.6 × 10-19Joules
- 1 kW•hr = 3.6 × 106 Joules = 2.25 × 1025 eV
- 1 MeV = 106eV, 1 GeV= 109eV, 1 TeV = 1012eV
- mass - eV/c2
- 1 eV/c2 = 1.78 × 10-36kg
- electron mass = 0.511 MeV/c2
- proton mass = 938 MeV/c2 = 0.938 GeV/ c2
- neutron mass = 939.6 MeV/c2
- momentum - eV/c:
- 1 eV/c = 5.3 × 10-28kg m/s
- momentum of baseball at 80 mi/hr 5.29 kgm/s 9.9×1027eV/c
- Distance
- 1 femtometer (“Fermi”) = 10-15 m

Radioactivity

- Discovery of Radioactivity
- Antoine Becquerel (1896): serendipitous discovery of radioactivity: penetrating radiation emitted by substances containing uranium
- Antoine Becquerel, Marie 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)
- Beta (b) rays have same q/m ratio as electrons
- Alpha (a) rays have same q/m ratio as He nucleus
- Alpha (a) rays captured in container show He-like emission spectrum

Geiger, Marsden, Rutherford expt.

- (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
- measure scattering angles of particles with scintillating screen (ZnS)

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

1871-1937

Proton

- “Canal rays”
- 1898: Wilhelm Wien:

opposite of “cathode rays”

- Positive charge in

nucleus (1900 – 1920)

- Atoms are neutral
- positive charge needed to cancel electron’s negative charge
- Rutherford atom: positive charge in nucleus
- periodic table realized that the positive charge of any nucleus could be accounted for by an integer number of hydrogen nuclei -- protons

Neutron

- Walther Bothe 1930:
- bombard light elements (e.g. 49Be) with alpha particles neutral radiation emitted
- Irène and Frédéric 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

Beta decay -- neutrino

- Beta decay puzzle :
- decay changes a neutron into a proton
- apparent “non-conservation” of energy
- apparent non-conservation of angular momentum
- Wolfgang Pauli predicted a light, neutral, feebly interacting particle (called it neutron, later called neutrino by Fermi)
- Although accepted since it “fit” so well, not actually observed initiating interactions until 1956-1958 (Cowan and Reines)

Puzzle with Beta Spectrum

- Three-types of radioactivity: a, b, g
- Both a, g discrete spectrum because

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)

Pauli’s neutrino letter

Dear Radioactive Ladies and Gentlemen!

I have hit upon a desperate remedy to save…the law of conservation of energy.

…there could exist electrically neutral particles, which I will call neutrons, in the nuclei…

The continuous beta spectrum would then make sense with the assumption that in beta decay, in addition to the electron, a neutron is emitted such that the sum of the energies of neutron and electron is constant.

But so far I do not dare to publish anything about this idea, and trustfully turn first to you, dear radioactive ones, with the question of how likely it is to find experimental evidence for such a neutron…

I admit that my remedy may seem almost improbable because one probably would have seen those neutrons, if they exist, for a long time. But nothing ventured, nothing gained…

Thus, dear radioactive ones, scrutinize and judge.

http://www.symmetrymagazine.org/cms/?pid=1000450

Positron

- Positron (anti-electron)
- Predicted by Dirac (1928) -- needed for relativistic quantum mechanics
- existence of antiparticles doubled the number of known particles!!!

Positron track going

upward through lead

plate

- P.A.M. Dirac
- Nobel Prize (1933)
- member of FSU faculty (1972-1984)
- one of the greatest physicists of the 20th century

Structure of nucleus

- size (Rutherford 1910, Hofstadter 1950s):
- 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)
- spin ½ Pauli principle only two identical nucleons per energylevel

Nuclear Sizes - examples

Find the ratio of the radii for the following nuclei:

1H, 12C, 56Fe, 208Pb, 238U

1 : 2.89 : 3.83 : 5.92 : 6.20

A, N, Z

- for natural nuclei:
- Z range 1 (hydrogen) to 92 (Uranium)
- A range from 1 ((hydrogen) to 238 (Uranium)
- N = neutron number = A-Z
- N – Z = “neutron excess”; increases with Z
- nomenclature:
- ZAXN or AXN orAX or X-A

Atomic mass unit

- “atomic number” Z
- Number of protons in nucleus
- 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

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 = 939.57 MeV/c2 =1.008665 u = 1839 me
- spin ½, magnetic moment -1.9 eħ/2mn

Nuclear masses, isotopes

- Nuclear masses measured, e.g. by mass spectrography
- masses expressed in atomic mass units (amu),

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)
- U- 235 (0.7% of natural U), U-238 (99.3% of natural U),

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

(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

Nuclear Masses, binding energy

- Mass of Nucleus Z(mp) + N(mn)
- “mass defect” m = difference between mass of nucleus and mass of constituents
- energy defect = binding energy EBEB = mc2
- binding energy = amount of energy that must be invested to break up nucleus into its constituents
- binding energy per nucleon = EB /A

Nuclear Binding Energy

- Nuclear binding energy = 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

Problem – set 4

- Compute binding energy per nucleon for
- 42He 4.00153 amu
- 168O 15.991 amu
- 5626Fe 55.922 amu
- 23592U 234.995 amu
- Is there a trend?
- If there is, what might be its significance?
- note:
- 1 amu = 931.5 MeV/c2
- m(proton) = 1.00782 amu
- m(neutron)= 1.00867 amu

Nuclear Radioactivity

- Alpha Decay
- AZ A-4(Z-2) + 4He
- Number of protons is conserved.
- Number of neutrons is conserved.
- Gamma Decay
- AZ* AZ +
- An excited nucleus loses energy by emitting a photon.

Beta Decay

- Beta Decay
- 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
- A proton and an electron have converted into a neutron and a neutrino.

Law of radioactive decay

- Activity A = number of decays per unit time
- decay constant = probability of decay per unit time
- Rate of decay number N of nuclei
- Solution of diff. equation (N0 = nb. of nuclei at t=0)
- Mean life = 1/

Nuclear decay rates

At t = 1/, N is 1/e (0.368) of the original amount

Nuclear (“strong”) force

- 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
- strong force must have very short range – << atomic size, otherwise isotopes would not have same chemical properties

Strong force -- 2

- 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

N-Z and binding energy vs A

- 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
- N-Z must be bigger for heavy nuclei (neutrons provide attraction without electrostatic repulsion
- heaviest stable nucleus: 209Bi – all nuclei heavier than 209Bi are unstable (radioactive)

Nuclear Models – liquid drop model

- 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

Bethe – Weizsäcker formula for binding energy

- 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
- Coulomb energy (electrostatic repulsion of the protons,)
- an asymmetry term (N vs Z)
- an exchange (pairing) term (even-even vs odd-even vs odd-odd number of nucleons)

Independent Particle Models

- 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”)
- derives “magic numbers” (of protons and/or neutrons) for which nuclei are particularly stable: 2, 8, 20, 28, 50, 82, 126, ..

Ground State

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

Fermi-Gas Model of NucleusPotential well

Collective model

- 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

Nuclear energy

- very heavy nuclei:
- energy released if break up into two medium sized nuclei
- “fission”
- light nuclei:
- energy released if two light nuclei combine -- “fuse” into a heavier nucleus – “fusion”

A, N, Z

- for natural nuclei:
- Z range 1 (hydrogen) to 92 (Uranium)
- A range from 1 ((hydrogen) to 238 (Uranium)
- N = neutron number = A-Z
- N – Z = “neutron excess”; increases with Z
- nomenclature:
- ZAXN or AXN orAX or X-A

Nuclear Energy - Fission

+ about 200 MeV energy

Sun’s Power Output

- Unit of Power
- 1 Watt = 1 Joule/second
- 100 Watt light bulb = 100 Joules/second
- Sun’s power output
- 3.826 x 1026 Watts
- exercise: calculate sun’s power output using Stefan-Boltzmann law (assume sun is a black body)

The Proton-Proton Cycle

1H + 1H → 2H + e+ + n

e+ + e- → g + g

2H + 1H → 3He + g

3He + 3He → 4He + 1H + 1H

1 pp collision in 1022→ fusion!

4H →4He

Deuterium creation

3He creation

4He creation

Life of a 20 Solar Mass Super-Giant

- Hydrogen fusion
- ~ 10 million years
- Helium fusion
- ~ 1 million years
- Carbon fusion
- ~ 300 years
- Oxygen fusion
- ~ 9 months
- Silicon fusion
- ~ 2 days

http://cassfos02.ucsd.edu/public/tutorial/SN.html

Carbon fusion

7.65 MeV above 12C ground state

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 nuclei 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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