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Nuclear and Particle Physics 3 lectures:

Nuclear Physics Particle Physics 1 Particle Physics 2

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Nuclear Physics Topics

Composition of Nucleus features of nuclei Nuclear Models nuclear energy

Fission Fusion

Summary

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About Units Energy - 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/c

Distanceo 1 femtometer (“Fermi”) = 10-15 m

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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):

o also other heavy elements (thorium, radium) show radioactivity

o 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)

o three kinds of rays have different electrical charge: : +, : -, : 0a b g

Identification of radiation: Ernest Rutherford (1899)

o Beta (b) rays have same q/m ratio as electrons o Alpha (a) rays have same q/m ratio as He nucleuso Alpha (a) rays captured in container show He-like emission

spectrum

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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)

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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 fromscattering 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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Proton “Canal rays”

1898: Wilhelm Wien: opposite of “cathode rays”

Positive charge in nucleus (1900 – 1920)Atoms are neutral

o positive charge needed to cancel electron’s negative charge

o Rutherford atom: positive charge in nucleusperiodic 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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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: (2

4He++) + 49Be 6

13C*

613C* 6

12C + n

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Beta decay -- neutrino 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 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)

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Desperate Idea of Pauli

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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

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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 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 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 energy level

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Nuclear Sizes - examples

)(Ar r 3

1

o ro = 1.2 x 10-15 m

Find the ratio of the radii for the following nuclei:

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

3

1

3

1

3

1

3

1

3

1

238:208:56:12:1

1 : 2.89 : 3.83 : 5.92 : 6.20

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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 or

AX or X-A

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Atomic 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 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

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Properties 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 me

spin ½, magnetic moment 2.79 eħ/2mp

NeutronCharge = 0Mass = 1.675 x 10-27 kg = 939.57 MeV/c2

= 1.008665 u = 1839 me

spin ½, magnetic moment -1.9 eħ/2mn

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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),

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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

(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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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 EB EB = mc2

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

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

1 amu = 931.49 MeVm(proton) 1.00782m(neutron) 1.00867

A = 56Z = 26N = 30

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

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Problem – set 4

Compute binding energy per nucleon for 4

2He 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 amum(neutron)= 1.00867 amu

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Binding energy per nucleon

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Nuclear Radioactivity

Alpha DecayAZ A-4(Z-2) + 4He

oNumber of protons is conserved.oNumber of neutrons is conserved.

Gamma DecayAZ* AZ +

oAn excited nucleus loses energy by emitting a photon.

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Beta Decay

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

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

Positron Decay AZ 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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Radioactivity

Several decay processes:

a decay:

b- decay:

b+ decay:

HePbPoge

HeYX AZ

AZ

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20682

21084

42

42

.,.

~9944

9943

~

1

.,.

eRbTcge

eYX AZ

AZ

eCNge

eYX AZ

AZ

126

127

1

.,.

Electron capture:

g decay:

CeNge

YeX AZ

AZ

126

127

1

.,.

)140(.,. 9943

*9943

*

keVTcTcge

XX AZ

AZ

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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/

dNA

dt

dNN

dt

1

0

0

dte

dtet

dN

dNt

t

t

0( ) tN t N e

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Nuclear 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)

Nu

cle

i R

em

ain

ing

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

0( ) tN t N e

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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

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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

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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)

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EB/A vs A

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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

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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:

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

NZa

A

ZaAaAa)Z,A(B λ

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“liquid drop” terms in B-W formula

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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, ..

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Fermi-Gas Model of Nucleus

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

Potential well

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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

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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”

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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 or

AX or X-A

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Nuclear Energy - Fission

+ about 200 MeV energy

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Fission

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Nuclear Fusion

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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)

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The Proton-Proton Cycle1H + 1H → 2H + e+ + n

e+ + e- → g + g2H + 1H → 3He + g

3He + 3He → 4He + 1H + 1H

Deuterium creation 3He creation 4He creation

4H → 4He

1 pp collision in 1022 → fusion!

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Super Kamiokande: Solar Neutrinos

Solar neutrino

Electron

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A Nearby Super-Giant

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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

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Carbon fusion

7.65 MeV above 12C ground state

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Oxygen fusion

7.12 MeV7.19 MeV

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Supernova 1987A Before

After

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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