Nuclear and ParticlePhysics

3rd Year Junior HonoursCourse

Dr Daniel Watts

Lecture 6

Main points of lecture 5

Pion exchange

attractiverepulsive

equilibriumposition

distance

nucleo

n-nu

cleo

n P.

E.

π

ω

Heavier meson exchange

V = Vc + VS + VT + VLS + .. Nuclear force is complicated !

Binding energy is a very important quantity in nuclear physics - Tells you about:

1) Forces between nucleons2) Stability of nuclei3) Energy released (or

required) in nuclear decays and reactions.

(

Binding energy per nucleon representsthe average bindingfelt by a nucleon in the nucleus

Maximum near to 56Fe

Notes Notes

The nucleus and its structure

Presently no complete theory to fully describe structure and behaviour of nuclei based solely on knowledge of force between nucleons(Although tremendous progress in past few years for A<12 !)

∴use MODELS:• simplifying assumptions• give reasonable account of observed properties• make predictions

LIQUID-DROP MODELnucleus regarded as collection of neutrons and protonsforming a droplet of incompressible fluid

⇒ good description of overall trend of binding energy per nucleon⇒ fails to account for magic numbers or give any prediction for Jπ

SHELL MODELneutrons and protons arranged in stable quantum statesin common potential well

⇒ accounts for ground-state properties (e.g. Jπ ) and magic numbers⇒ does not predict many of the observed nuclear excited states

COLLECTIVE MODELSneutrons and protons show collective motions give rise to vibrational and rotational states

⇒ accounts for properties of non-spherical nuclei⇒ fails to reproduce other features

How to explain these properties?

LIQUID DROP MODELnucleus regarded as collection of neutrons and protons

forming a droplet of incompressible fluid

Bethe and von Weizsäcker (1935):

Semi Empirical Mass Formula (SEMF)Remember

av = 15.6 MeVas = 17.2 MeVaa = 23.3 MeVaC = 0.70 MeV

From fits to experimental binding energies:

Contributions from various terms

δ > 0 for even N – even Zδ < 0 for odd N – odd Zδ = 0 for odd A

pairing term

3/13/2

AZaAaAa)Z,A(B

2

Csv −−= + δ (A,Z)A

)Z2A(a2

a−−

Notes Notes

Aav

3/1

2

AZaC−

AZAaa

2)2( −−

3/2Aas−

+ δ (A,Z)

Volume energy: Each nucleon only feels interaction of close neighbours due to short range of nuclear force – Gives a positive binding energy which is roughly the same for each nucleon

Surface correction: Nucleons near surface of nucleus surrounded by fewer nucleons and will therefore experience less attractive potential energy than those inside the nucleus. Compensate with a reduction in binding energy proportional to number of nucleons in nuclear surface

Coulomb energy: Nucleus has total charge Ze confined to a sphere of radius R. The resultant potential energy given by electrostatic theory

Symmetry term:Stable light nuclei haveN~Z (i.e. A~2Z). If A deviates from 2Z then binding energy is reduced.

Pairing term: Most stable nuclei have Z even and N even and therefore A even (even-even nuclei). Increases binding for even-even nuclei and reduces for odd-odd.

RZe

0

2

4

)(

5

3

πε=

M(A,Z)c2 = aA + bZ + cZ2 ± δ (A,Z)

… more on nuclear masses

quadratic function in Z ⇒ PARABOLA

odd A even A

Minimum stable isobar ⇒ 0Zm

.constA

=∂∂

=

3/2stable A015.0972.1AZ

+=

closest integer number to Zstable gives atomic number for stable configuration at given A

odd A ⇒ δ = 0 ⇒ one parabola only even A ⇒ ± δ ⇒ two parabolae

Notes Notes

LIQUID DROP MODEL

Accounts for:

• overall trend of binding energies• correctly predicts masses of most nuclei

Does not account for:

• extra binding at MAGIC NUMBERSZ, N = 2, 8, 20, 28, 50, 82, (126)

• abundance peaks at magic numbers• even-even nuclei more stable than odd-odd nuclei• discontinuities in neutron binding energy vs. A• ground state spin and parity properties of even A nuclei• energies of low lying excited states (higher than expected)

SHELL MODEL

need for refined model to explain observational facts

Fission and FusionCurve of binding energy per nucleon

Fission Process where a heavy nucleus splits into lighter fragments

Energy is released when B/A for the initial nucleus is smaller than for the final nuclei.

Lighter nuclei have smaller B/A because of their smaller coulomb repulsion energy

Example: Nucleus with A=200 fissions into 2 nuclei with A=100A=200 B/A ~ 7.6 MeV/nucleonA=1 00 B/A ~ 8.5 MeV/nucleon B/A increases by 8.5-7.6 = 0.9 MeV Total binding energy increases by ~200(0.9) = 180MeV !!

⇒ A lot of energy is released in each fission reaction. Energy goes into the kinetic energies of fission fragments, neutrons …

Variations in binding energy per nucleon have verysignificant effects → result in large energy releases in nuclearreactions.

Underpins energy generation in stars, power from fission

FissionFusio

n

Pote

ntia

l Ene

rgy

Distance

Activation energy

Activation energydecreases asA increasesGives upper limit to size of nuclei

Notes Notes

Neutrons likely to be released in fission as fewer excess neutrons necessary to counteract the coulomb repulsion in the lighter fragments

These neutrons can be used to set up a chain reaction with the release of large amounts of energy.

FissionFusion - combining nuclei to form heavier nucleus with higher B/A

B/A larger in the fused nucleus as a smaller fraction of nucleonsare near the surface of the nucleus - therefore the average strong force felt by nucleons is larger (i.e. surface term in SEMF)

Fusion is the energy source of stars. Gravitational potential energy converted into kinetic & radiation energy of gas → Temperature of gas rises & fusion ignitesNuclear power keeps star interior hot enough to stop the collapse(at least for a while! e.g ~9 billion years in the case of our Sun)

Fusion

MeVeDpp e 42.021

11

11 +++→+ +ν

MeVHepD 49.532

11

21 ++→+ γ

MeVppHeHeHe 9.1211

11

42

32

32 +++→+Get energy release of 26.7MeV per 4He nucleus formed

First step is rare (involves weak interaction)Sets long timescalesFor stellar life

Notes Notes

Important quantities

B(A,Z)=∆Mc2 = [ZmH + Nmn]c2 - Ma(A,Z)c2 Binding energy

Sn(A,Z) = [M(A-1,Z) – M(A,Z) + Mn ]c2 Neutron separation energy

Sp(A,Z) = [ M(A-1,Z-1) – M(A,Z) + Mp ]c2 Proton separation energy

M(A,Z)= [ZmH + Nmn] - B(A,Z)/c2 Nuclear mass

Proton (neutron) separation energies correspond to the binding energy of the last proton (neutron)

Tutorial 3:

1) How much energy is necessary to split up an alpha particle into its constituent nucleons?(Mn=1.008665u, MH=1.007825u, M(4He)=4.00260u).

2 How much energy is required to remove a proton from 56Fe(Z=26). Also calculate the binding energy per nucleon for 56Fe and compare with the proton separation energy. Can you explain any differences?( M(56Fe) = 55.934939u, M(55Mn) = 54.938047u)

3) The atomic mass of a nucleus is predicted by the semi-empirical mass formula to be given by

M(N,Z)c2 = ZMHc2 + NMn c2 - avA + asA2/3 + acZ

2/A1/3 + asym(N-Z)2/A + δ

Where MH = 938.8 MeV/c2, Mn = 939.6 MeV/c2, av = 15.8 MeV, as= 18.0 MeV, ac = 0.7 MeV and asym = 25.3 MeV. Mirror nuclei are nuclei with the same mass number A and interchanged values of N and Z. Which terms in the above formula give rise to differences in atomic mass between a pair of mirror nuclei? Calculate the predicted atomic mass difference between 11C(Z=6) and 11B(Z=5) in units of MeV/c2. From this information and any general considerations, what can you deduce about the possible decay modes for these two nuclei?

4) In the semi-empirical mass formula what experimental evidence does the pairing term δ account for? For a given value of the mass number A, nuclear masses can be expressed as a quadratic function in Z. Using the“minimum isobar” formula (as derived in the lecture notes) determine whether the 142Xe(Z=54) nucleus is β+ or β- unstable.

5) Specify the spins and parities of nuclear energy levels to which a d-wave neutron transition can take place from the following states

a) Jπ=0+ b) 1-