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Nuclear and Particle PhysicsNuclear and Particle Physics

3 lectures:Nuclear Physics

Particle Physics 1Particle Physics 2

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Nuclear Physics TopicsNuclear Physics Topics

Composition of Nucleusfeatures of nucleiNuclear Modelsnuclear energy

FissionFusion

Summary

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About UnitsAbout UnitsEnergy - electron-volt

1 electron-volt = kinetic energy of an electron when moving through potential difference of 1 Volt;

o 1 eV = 1.6 × 10-19 Jouleso 1 kW•hr = 3.6 × 106 Joules = 2.25 × 1025 eVo 1 MeV = 106 eV, 1 GeV = 109 eV, 1 TeV = 1012 eV

mass - eV/c2

o 1 eV/c2 = 1.78 × 10-36 kgo electron mass = 0.511 MeV/c2

o proton mass = 938 MeV/c2 = 0.938 GeV/ c2

o neutron mass = 939.6 MeV/c2

momentum - eV/c: o 1 eV/c = 5.3 × 10-28 kg m/so momentum of baseball at 80 mi/hr ≈ 5.29 kgm/s ≈ 9.9 × 1027

eV/cDistance

o 1 femtometer (“Fermi”) = 10-15 m

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RadioactivityRadioactivityDiscovery of Radioactivity

Antoine Becquerel (1896): serendipitous discovery of radioactivity: penetrating radiation emitted by substances containing uraniumA. Becquerel, Maria Curie, Pierre Curie(1896 – 1898):

o also other heavy elements (thorium, radium) show radioactivityo three kinds of radiation, with different penetrating power

(i.e. amount of material necessary to attenuate beam): “Alpha (α) rays” (least penetrating – stopped by paper)“Beta (β) rays” (need 2mm lead to absorb) “Gamma (γ) rays” (need several cm of lead to attenuated)

o three kinds of rays have different electrical charge: α: +, β: −, γ: 0

Identification of radiation:Ernest Rutherford (1899)

o Beta (β) rays have same q/m ratio as electrons o Alpha (α) rays have same q/m ratio as Heo Alpha (α) rays captured in container show He-like emission spectrum

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Geiger, Geiger, MarsdenMarsden, Rutherford , Rutherford exptexpt..(Geiger, Marsden, 1906 - 1911) (interpreted by Rutherford, 1911)get α 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 Geiger MarsdenMarsden experiment: resultexperiment: 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 Rutherford1871-1937

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ProtonProton“Canal rays”

1898: Wilhelm Wien: opposite of “cathode rays”

Positive charge in nucleus (1900 – 1920)

Atoms are neutralo positive charge needed to cancel electron’s negative chargeo 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

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NeutronNeutronWalther Bothe 1930:

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 -- puzzlepuzzle 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 1920observed reaction was: α (24He++) + 49Be → 6

13C*6

13C* → 612C + n

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Beta decay Beta decay ---- neutrinoneutrino

Beta decay puzzle : o decay changes a neutron into a proton o apparent “non-conservation” of energyo 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)

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Puzzle with Beta SpectrumPuzzle with Beta Spectrum

Three-types of radioactivity: α, β, γBoth α, γ discrete spectrum because

Eα, γ = Ei – EfBut β 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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Desperate Idea of Desperate Idea of PauliPauli

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PositronPositronPositron (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 leadplate

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

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Structure of nucleusStructure of nucleussize (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 levelstwo distinct (but similar) sets of energy levels, one for protons, one for neutronsproton energy levels slightly higher than those of

neutrons (electrostatic repulsion)spin ½ ⇒ Pauli principle ⇒ only two identical nucleons per eng. level

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Nuclear Sizes Nuclear Sizes -- examplesexamples

)(Ar r 31

o= ro = 1.2 x 10-15 m

Find the ratio of the radii for the following nuclei:

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

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31

31

31

31

238:208:56:12:1

1 : 2.89 : 3.83 : 5.92 : 6.20

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A, N, ZA, N, Z

for natural nuclei: Z range 1 (hydrogen) to

92 (Uranium)A range from 1 ((hydrogen)

to 238 (Uranium)N = neutron number = A-ZN – Z = “neutron excess”;

increases with Znomenclature:

ZAXN or AXN orAX or X-A

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Atomic mass unitAtomic mass unit

“atomic number” ZNumber of protons in nucleus

Mass Number ANumber of protons and neutrons in nucleus Atomic mass unit is defined in terms of the mass of 1212

66CC, with A = 12, Z = 6: 1 amu = (mass of 1212

66CC atom)/12 1 amu = 1.66 x 10-27kg 1 amu = 931.494 MeV/c2

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Properties of NucleonsProperties of Nucleons

Proton Charge = 1 elementary charge e = 1.602 x 10-19 CMass = 1.673 x 10-27 kg = 938.27 MeV/c2 =1.007825 u = 1836 mespin ½, magnetic moment 2.79 eħ/2mp

NeutronCharge = 0Mass = 1.675 x 10-27 kg = 939.6 MeV/c2 = 1.008665 u = 1839 mespin ½, magnetic moment -1.9 eħ/2mn

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Nuclear masses, isotopesNuclear masses, isotopes

Nuclear masses measured, e.g. by mass spectrographymasses 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 neutronsexamples:

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),

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Nuclear energy levels: exampleNuclear 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

Δx Δp = h/2π Δx Δ(cp) = hc/2π

Δ(cp) = hc/(2π Δx) = hc/(2π r)

Δ(cp) = 6.63x10-34 Js * 3x108 m/s / (2π * 1.33x10-15 m)

Δ(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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Nuclear Masses, binding energyNuclear Masses, binding energyMass of Nucleus < Z(mp) + N(mn)

“mass defect” Δm = difference between mass of nucleus and mass of constituentsenergy defect = binding energy EB

EB = Δm c2

binding energy = amount of energy that must be invested to break up nucleus into its constituentsbinding energy per nucleon = EB /A

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Nuclear Binding EnergyNuclear Binding Energy

The energy (or mass) difference between the nucleus and its constituent neutrons and protons.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

1 amu = 931.5 MeVm(proton) 1.00782m(neutron) 1.00867

A = 56Z = 26N = 30

Mass (amu) 55.92066Ebinding -505.58094EB/A -9.02823

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ProblemProblem

Compute binding energy per nucleon for 4

2He 4.00153 amu16

8O 15.991 amu56

26Fe 55.922 amu235

92U 234.995 amuIs there a trend?If there is what might be its significance?

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Binding energy per nucleonBinding energy per nucleon

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Nuclear RadioactivityNuclear Radioactivity

Alpha DecayAZ A-4(Z-2) + 4Heo Number of protons is conserved.o Number of neutrons is conserved.

Gamma DecayAZ* AZ + γo An excited nucleus loses energy by emitting

a photon.

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Beta DecayBeta Decay

Beta DecayAZ A(Z+1) + e- + an anti-neutrino

o A neutron has converted into a proton, electron and an anti-neutrino.

Positron DecayAZ A(Z-1) + e+ + a neutrino

o A proton has converted into a neutron, positron and a neutrino.

Electron Capture AZ + e- A(Z-1) + a neutrino

o A proton and an electron have converted into a neutron and a neutrino.

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RadioactivityRadioactivity

Several decay processes:α decay:

β− decay:

β+ decay:

HePbPoge

HeYX AZ

AZ

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20682

21084

42

42

.,. +→

+→ −−

~9944

9943

~

1

.,. ν

ν

++→

++→

−

−+

eRbTcge

eYX AZ

AZ

ν

ν

++→

++→

+

+−

eCNge

eYX AZ

AZ

126

127

1

.,.

Electron capture:

γ decay:ν

ν

+→+

+→+

−

−−

CeNge

YeX AZ

AZ

126

127

1

.,.

)140(.,. 9943

*9943

*

keVTcTcge

XX AZ

AZ

γ

γ

+→

+→

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Law of radioactive decayLaw of radioactive decay

Activity A = number of decays per unit timedecay constant λ =

probability of decay per unit timeRate of decay ∝ number N of nuclei Solution of diff. equation (N0 = nb. of nuclei at t=0)Mean life τ = 1/ λ

.dtdNA =

.NdtdN λ−=

λτ

λ

λ

1

0

0 ===

∫

∫

∫∫

∞−

∞−

dte

dtet

dN

dNt

t

t

.)( 0teNtN λ−=

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Nuclear decay ratesNuclear decay rates

Nuclear Decay

0.0

200.0

400.0

600.0

800.0

1000.0

0.0 1.0 2.0 3.0 4.0 5.0

Time(s)

Nuc

lei R

emai

ning

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

.)( 0teNtN λ−=

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Nuclear (Nuclear (““strongstrong””) force) force

atomic nuclei small -- about 1 to 8fmat 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 energiesmirror 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 Strong force ---- 22

range: fades away at distance ≈ 3fmforce between 2 nucleons at 2fm distance ≈

2000Nnucleons on one side of U nucleus hardly

affected by nucleons on other sideexperimental evidence for nuclear force from

scattering experiments; low energy p or α scattering: scattered particles

unaffected by nuclear forcehigh energy p or α scattering: particles can overcome electrostatic repulsion and can penetrate deep enough to enter range of nuclear force

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NN--Z and binding energy Z and binding energy vsvs AAsmall nuclei (A<10):

All nucleons within range of strong force exerted by all other nucleons;add another nucleon ⇒ enhance overall cohesive force ⇒ EB rises

sharply with increase in Amedium size nuclei (10 < A < 60)

nucleons on one side 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 attractionRepulsive electrostatic forces (infinite range!) begin to have

stronger effect N-Z must be bigger for heavy nuclei (neutrons provide attraction

without electrostatic repulsionheaviest stable nucleus: 209Bi – all heavier nuclei unstable

(radioactive)

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EEBB/A /A vsvs AA

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Nuclear Models Nuclear Models –– liquid drop modelliquid 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

nucleipredicts 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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BetheBethe –– WeizsWeizsääckercker formula for binding energyformula 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:

o volume energyo surface energyo Coulomb energy (electrostatic repulsion of the protons,)o an asymmetry term (N vs Z)o an exchange (pairing) term (even-even vs odd-even vs odd-odd

number of nucleons)

( ) 3/4-P

2

Sym1/3

2

C3/2

SV A a A

NZaAZaAaAa)Z,A(B λ−

−−−−=

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““liquid drop liquid drop ”” terms in Bterms in B--W formulaW formula

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Independent Particle ModelsIndependent Particle Models

assume nucleons move inside nucleus without interacting with each otherFermi- gas model:

Protons and neutrons move freely within nuclear volume, considered a rectangular boxProtons and neutrons are distinguishable and so move in separatepotential wells

Shell Model formulated (independently) by Hans Jensen and Maria Goeppert-MayerEach 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”)derive “magic numbers” (of protons and/or neutrons) for which

nuclei are particularly stable: 2, 8, 20, 28, 50, 82, 126, ..

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FermiFermi--Gas Model of NucleusGas Model of Nucleus

Ground StateIn 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

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Collective modelCollective 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 coreadditional nucleons move in potential well due to

interaction with the coreinteraction of external nucleons with the core ⇒

agitate core – set up rotational and vibrationalmotions in core, similar to those that occur in dropletsgives best quantitative description of nuclei

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Nuclear energyNuclear energyvery 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”

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Nuclear Energy Nuclear Energy -- FissionFission

+ about 200 MeV energy

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FissionFission

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Nuclear FusionNuclear Fusion

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SunSun’’s Power Outputs Power Output

Unit of Power1 Watt = 1 Joule/second100 Watt light bulb = 100 Joules/second

Sun’s power output3.826 x 103.826 x 102626 WattsWatts

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The ProtonThe Proton--Proton CycleProton Cycle 1H + 1H → 2H + e+ + νe+ + e- → γγ + γγ

2H + 1H → 3He + γγ

3He + 3He → 4He + 1H + 1H

Deuterium creation 3He creation 4He creation

4H → 4He

1 pp collision in 1022 → fusion!

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Super Super KamiokandeKamiokande: Solar Neutrinos: Solar Neutrinos

Solar neutrino

Electron

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A Nearby SuperA Nearby Super--GiantGiant

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Life of a 20 Solar Mass SuperLife of a 20 Solar Mass Super--GiantGiant

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

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Supernova 1987ASupernova 1987A Before

After

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StardustStardust

7.65 MeV above 12C ground state

Sir Fred Hoyle1915-2001

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Stardust Stardust –– IIII

7.12 MeV7.19 MeV

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SummarySummary

nuclei made of protons and neutrons, held together by short-range strong nuclear forcemodels describe most observed features, still

being tweaked and modified to incorporate newstobservationsno full-fledged theory of nucleons yetdevelopment of nuclear theory based on QCD has

begun nuclear fusion is the process of energy production

of Sun and other starswe (solar system with all that’s in it) are made of

debris from dying stars