7-1

CHEM 312 Lecture 7: Fission• Readings: Modern Nuclear Chemistry, Chapter

11; Nuclear and Radiochemistry, Chapter 3 • General Overview of Fission• Energetics• The Probability of Fission• Fission Product Distributions

§ Total Kinetic Energy Release§ Fission Product Mass Distributions§ Fission Product Charge Distributions

• Fission in Reactors§ Delayed neutron

• Proton induced fission

7-2

Nuclear Fission• Fission discovered by Otto Hahn and Fritz

Strassman, Lisa Meitner in 1938§ Demonstrated neutron irradiation of

uranium resulted in products like Ba and Laà Chemical separation of fission products

• For induced fission, odd N§ Addition of neutron to form even N§ Pairing energy

• In 1940 G. N. Flerov reported that 238U undergoes fission spontaneously§ half life of round 1016 y§ Several other spontaneous fission isotopes

foundà Z > 90

§ Partial fission half lives from nanoseconds to 2E17 years

7-3

Fission• Can occur when enough energy is supplied by the bombarding particle for the

Coulomb barrier to be surmounted§ Fast neutron § Proton

• Spontaneous fission occurs by tunneling through barrier• Thermal neutron induces fission from pairing of unpaired neutron, energy gain

§ Nuclides with odd number of neutrons fissioned by thermal neutrons with large cross sections

§ follows1/v law at low energies, sharp resonances at high energies

7-4

Energetics Calculations

• Why does 235U undergo neutron induced fission for thermal energies?§ Where does energy come

from?• Generalized energy equation

§ AZ + n A+1Z + E• For 235U

§ E=(40.914+8.071)-42.441§ E=6.544 MeV

• For 238U§ E=(47.304+8.071)-50.569§ E=4.806 MeV

• For 233U§ E=(36.913+8.071)-38.141§ E=6.843 MeV

• Fission requires around 5-6 MeV

7-5

Fission Process• Nucleus absorbs energy

§ Excites and deforms § Configuration “transition state” or “saddle point”

• Nuclear Coulomb energy decreases during deformation§ Nuclear surface energy increases

• Saddle point key condition§ rate of change of the Coulomb energy is equal to the rate of change of the nuclear surface energy§ Induces instability that drives break up of nucleus

• If the nucleus deforms beyond this point it is committed to fission§ Neck between fragments disappears§ Nucleus divides into two fragments at the “scission point.”

à two highly charged, deformed fragments in contact • Large Coulomb repulsion accelerates fragments to 90% final kinetic energy within 10-20 s

7-6

Fission Products• Fission yield curve varies with fissile isotope• 2 peak areas for U and Pu thermal neutron induced fission• Variation in light fragment peak• Influence of neutron energy observed 235U fission yield

7-7

7-8

Fission• Primary fission products always on

neutron-excess side of stability§ high-Z elements that undergo

fission have much larger neutron-proton ratios than the stable nuclides in fission product region

§ primary product decays by series of successive - processes to its stable isobar

• Yields can be determined§ Independent yield: specific

for a nuclide § Cumulative yield: yield of

an isobarà Beta decay to valley of

stability§ Data for independent and

cumulative yields can be found or calculated

• For reactors§ Emission of several neutrons

per fission crucial for maintaining chain reaction

§ “Delayed neutron” emissions important in control of nuclear reactors

Comparison of cumulative and independent yields for A=141

http://www-nds.iaea.org/sgnucdat/c2.htm

7-9

Fission Process: Delayed Neutrons• Particles form more spherical shapes

§ Converting potential energy to emission of “prompt” neutrons

§ Gamma emission after neutrons§ Then decay

à Occasionally one of these decays populates a high lying excited state of a daughter that is unstable with respect to neutron emission

§ “delayed” neutrons§ 0.75 % of total neutrons from

fissionà 137-139I and 87-90Br as examples

7-10

Delayed Neutrons in Reactors• Control of fission

§ 0.1 msec for neutron from fission to reactà Need to have tight controlà 0.1 % increase per generation

* 1.001^100, 10 % increase in 10 msec

• Delayed neutrons useful in control§ Longer than 0.1 msec§ 0.65 % of neutrons delayed from 235U

à 0.26 % f• Power proportional to number of available

neutrons§ Should be kept constant under changing

conditionsà Control elements and burnable

poisons§ k=1 (multiplication factor)

à Ratio of fissions from one generation to the next

* k>1 at startupà or 233U and 0.21 % for 239Pu

7-11

Energetics• Any nucleus of A> 100 into two nuclei of

approximately equal size is exoergic. § Why fission at A>230

• Separation of a heavy nucleus into two positively charged fragments is hindered by Coulomb barrier§ Treat fission as barrier penetration

à Barrier height is difference between * Coulomb energy between the two

fragments when they are just touching* the energy released in the fission process

• Near uranium both these quantities have values close to 200 MeV

7-12

7-13

Energetics• Generalized Coulomb barrier equation

§ Compare with Q value for fission

• Determination of total kinetic energy§ Equation deviates at heavy actinides (Md, Fm)

• Consider fission of 238U§ Assume symmetric§ 238U119Pd + 119Pd + Q

à Z=46, A=119* Vc=462*1.440/(1.8(1191/3)2)=175 MeV* Q=47.3087-(2*-71.6203) = 190.54 MeV

§ asymmetric fission§ 238U91Br + 147La + Q

à Z=35, A=91à Z=57, A=147

* Vc=(35)(57)*1.44/(1.8*(911/3+1471/3))=164 MeV* Q=47.3087-(-61.5083+-66.8484) = 175.66 MeV

• Realistic case needs to consider shell effects

MeVAAZZ

RReZZ

c 3/12

3/11

21

21

221 96.0V

)(8.144.1V 3/1

23/1

1

21

AAZZ

c

7-14

Fission Isomers• Some isomeric states in heavy

nuclei decay by spontaneous fission with very short half lives§ Nano- to microseconds§ De-excite by fission process

rather than photon emission• Fissioning isomers are states in

these second potential wells § Also called shape isomers§ Exists because nuclear

shape different from that of the ground state

§ Proton distribution results in nucleus unstable to fission

• Around 30 fission isomers are known§ from U to Bk

• Can be induced by neutrons, protons, deuterons, and a particles§ Can also result from decay

7-15

Fission Isomers: Double-humped fission barrier•At lower mass numbers, the second barrier is rate-determining, whereas at larger A, inner barrier is rate determining•Symmetric shapes are the most stable at two potential minima and the first saddle, but some asymmetry lowers second saddle

7-16

Proton induced fission

• Energetics impact fragment distribution

• excitation energy of the fissioning system increases§ Influence of ground

state shell structure of fragments would decrease

§ Fission mass distributions shows increase in symmetric fission

7-17

Topic Review

• Mechanisms of fission§ What is occurring in the nucleus during fission

• Understand the types of fission§ Particle induced§ Spontaneous

• Energetics of fission§ Q value and coulomb barrier

• The Probability of Fission§ Cumulative and specific yields

• Fission Product Distributions§ Total Kinetic Energy Release§ Fission Product Mass Distributions

7-18

Questions

• Compare energy values for the symmetric and asymmetric fission of 242Am.

• What is the difference between prompt and delayed neutrons in fission.

• What is the difference between induced and spontaneous fission.

• What influences fission product distribution?• Compare the Coulomb barrier and Q values for the fission

of Pb, Th, Pu, and Cm.• Describe what occurs in the nucleus during fission.• Compare the energy from the addition of a neutron to

242Am and 241Am. Which isotope is likely to fission from an additional neutron.

7-19

Pop Quiz

• Provide calculations showing why 239Pu can be fissioned by thermal neutron but not 240Pu.

• Compare the Q value and Coulomb energy (Vc) from the fission of 239Pu resulting in 138Ba and 101Sr. Is this energetically favored?

• Provide comments on blog• Bring to class on 17 October

7-20

CHEM 312 Lecture 8: Nuclear Force, Structure and Models

• Readings:§ Nuclear and

Radiochemistry: Chapter 10 (Nuclear Models)

§ Modern Nuclear Chemistry: Chapter 5 (Nuclear Forces) and Chapter 6 (Nuclear Structure)

• Characterization of strong force

• Charge Independence § Introduce isospin

• Nuclear Potentials• Simple Shell Model (Focus of

lecture)• Nucleus as a Fermi Gas

Nuclear Force• For structure, reactions and decay of

nuclei§ electromagnetic, strong and weak

interactions are utilized• Fundamental forces exhibit exchange

character§ operate through virtual exchange

of particles that act as force carriers

7-21

Charge Independent Force• Strong force not effected by

charge§ np, nn, pp interactions

the sameà Electromagnetic

force for charge• Strong force examined by

§ Nucleon-nucleon scattering

§ Mirror nucleià Isobars with number

of p in one nuclei equals number of n in other

à Similar energy for net nuclear binding energy* Normalize

influence of Coulomb Energy

• Shows proton and neutron two states of same particle

• Isospin is conserved in processes involving the strong interaction

• Isospin forms basis for selection rules for nuclear reactions and nuclear decay processes

• Property of nucleon§ Analogy to angular momentum§ T=1/2 for a nucleon

à +1/2 for proton, -1/2 for neutron

7-22

Shell Model• Model nucleus as a spherical rigid

container § square-well potential

à potential energy assumed to be zero when particle is inside walls

à Particle does not escape• Harmonic oscillator potential

§ parabolic shape§ steep sides that continue upwards

à useful only for the low-lying energy levels

à equally spaced energy levels* Potential does not

“saturate” * not suitable for large nuclei

• Change from harmonic oscillator to square well lowers potential energy near edge of nucleus§ Enhances stability of states near

edge of nucleus§ States with largest angular

momentum most stabilized

7-23

• Shell filling§ States defined by n and l

à 1s, 1p, 1d, …* Compare with electrons

§ States with same 2n+l degenerate with same parity (compose level)à 2s = 2*2+0=4à 1d = 2*1+2 =4à 1g=2*1+4=6à 2d=2*2+2=6à 3s=2*3+0=6

• Spin-Orbit Interaction§ Addition of spin orbit term causes energy

level separation according to total angular momentum (j=ℓ+s)à For p, l=1

* s=±1/2* j= 1+1/2=3/2 and 1-1/2=1/2* split into fourfold degenerate 1p3/2

and twofold degenerate 1p1/2 statesà For g, l=4, j=7/2 and 9/2

§ states with parallel coupling and larger total angular momentum values are favored

§ closed shells 28, 50, 82, and 126à splitting of the 1f, 1g, 1h, and 1i

• Each principal quantum number level is a shell of orbitals

• Energy gap between shell the same

Shell Model

7-24

Filling Shells• Odd-A Nuclei

§ In odd A nucleus of all but one of nucleons considered to have their angular momenta paired offà forming even-even coreà single odd nucleon moves essentially independently in this coreà net angular momentum of entire nucleus determined by quantum state of single odd

nucleon* Spin of spin of state, parity based on orbital angular momentum

Ø Even (s, d, g, i,….)Ø Odd (p, f, h,….)

• Configuration Interaction§ For nuclides with unpaired nucleons number half way between magic numbers nuclei single-

particle model is oversimplificationà Contribution from other nucleons in potential well, limitation of model

• Odd-Odd Nuclei§ one odd proton and one odd neutron each producing effect on nuclear moments§ No universal rule can be given to predict resultant ground state

• Level Order§ applied independently to neutrons and protons§ proton levels increasingly higher than neutron levels as Z increases

à Coulomb repulsion effect§ order given within each shell essentially schematic and may not represent exact order of filling

• Ground States of Nuclei§ filled shells spherically symmetric and have no spin or orbital angular momentum and no

magnetic moment§ ground states of all even-even nuclei have zero spin and even parity

à increased binding energy of nucleons

7-25

Filling Example• Consider isotope 7Li

§ 3 protons and 4 neutronsà 2 protons in 1s1/2, 1 proton in 1p3/2à 2 neutrons in 1s1/2, 2 neutrons in

1p3/2• spin and angular momentum based on unpaired

proton• spin should be 3/2 • nuclear parity should be negative

§ parity of a p-state (odd l value, l=1)• Excited state for 7Li?

§ Proton from 1p3/2 to 1p1/2à Breaking paired nucleons requires

significant energy, neutrons remain paired

§ Bound excited state corresponds to promotion of proton

§ 1p1/2 corresponds to 1/2-

7-26

Filling Levels•consider 13C

§ 7th neutron is unpaired§ p ½ state

à ½-•51V unpaired nucleon is 23rd proton, f 7/2

à 7/2-• Not always so straight forward

§ examine 137Baà 81st neutron is

unpaired, h 11/2à spin 11/2-à measured as 3/2+

•high spin does not appear as ground•Deformation impacts level filling

7-27

Shell Filling: Spin and parity for odd-odd nuclei

•Configurations with both odd proton and odd neutron have coupling rules to determine spin § Integer spin value

•Determine spin based on Nordheim number N §Nordheim number N (=j1+j2+ l1+ l2) is even, then

I=j1-j2• if N is odd, I=j1j2•Parity from sum of l states

§Even positive parity§Odd negative parity

•prediction for configurations in which there is combination of particles and holes is I=j1+j2-1•Examples on following page

7-28

Shell Model Example• Consider 38Cl

§ 17 protons (unpaired p in 1d3/2)à l=2 (d state) and j=3/2

§ 21 neutrons (unpaired n in 1f7/2)à l=3 (f state) and j=7/2à N= 2+3/2+3+7/2 = 10à Even; I=j1-j2à Spin = 7/2-3/2=2à Parity from l (3+2)=5

(odd), negative parity§ 2-

• Consider 26Al (13 each p and n)§ Hole in 1d5/2, each j = 5/2,

each l=2§ N=5/2+5/2+2+2=9§ N=odd; I=j1j2§ I = 0 or 5 (5 actual value)§ Parity 2+2=4, even, +§ 5+

7-29

Particle Model: Collective Motion in Nuclei• Effects of interactions not included in shell-model

description§ pairing force§ lack of spherically symmetric potential

• Nonspherical Potential§ intrinsic state

à most stable distribution of nucleons among available single-particle states

§ since energy require for deformation is finite, nuclei oscillate about their equilibrium shapesà Deformities 150 <A<190 and A<220

* vibrational levels§ nuclei with stable nonspherical shape have

distinguishable orientations in spaceà rotational levelsà polarization of even-even core by motion of odd

nucleon•Splitting of levels in shell model

§ Shell model for spherical nuclei•Deformation parameter e2

Prolate: polar axis greater than equatorial diameter

Oblate: polar axis shorter than diameter of equatorial circle

DR=major-minor axisProlate DR is positiveOblate DR is negative

7-30

Nilsson Diagram

• 50<Z<82•127I§ 53rd proton is

unpairedà7/2+ from

shell model§ measured as

5/2+•Deformation parameter should show 5/2, even l

§ Oblate nuclei

7-31

Fermi Gas Model

Potential energy well derived from the Fermi gas model. The highest filled energy levels reach up to the Fermi level of approximately 28 MeV. The nucleons are bound by approximately 8 MeV.

7-32

Review and Questions

• What is a nuclear potential• What are the concepts behind the following:

§ Shell model§ Fermi model

• How do nuclear shapes relate to quadrupole moments

• Utilize Nilsson diagrams to correlate spin and nuclear deformation

7-33

Pop Quiz

• Using the shell model determine the spin and parity of the following§ 19O§ 99Tc§ 156Tb§ 90Nb§ 242Am§ 4He

• Compare your results with the actual data. Which isotopes maybe non-spherical based on the results?

• Post comments on the blog• E-mail answers or bring to class