1

Nuclear Fission

• Through the 1930’s higher mass elements could be created by bombarding nuclei with neutrons, followed by beta decay

• Attempts to create transuranic elements failed, however. Instead, Barium and other lighter elements were identified in the reaction products.

• (1939) Meiner and Frisch proposed that Uranium undergoes fission, or splits into fragments, after neutron absorption

• Fission represents a competition between nuclear binding and Coulomb repulsion

Nuclear Binding ACoulomb Repulsion ~ Z2

04/19/23 Physics 590B - Fall 2014

2

Binding Energies

• Recall the binding energy per nucleon:

• A heavy nucleus like238U has B/A ~ 7.6 MeV/nucleon

• If 238U splits into two equal A=119 fragments, then B/A ~8.5 MeV/nucleon

• This would release E ~ 214MeVin the form of kinetic energyof the fragments

• Smaller fragments more energetically favorable

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3

Characteristics of Fission

• Consider the fission of 235U by thermal (low energy) neutrons:

– The fragments are not uniquely determined, but tend to favor unequal sizes

• Favored by phase spacearguments, both nucleicloser to stability

– “Fast” neutrons favor moreequal mass fragments

nCsRbnU 214193235 neutron rich

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Prompt and Delayed Neutrons

• What if we the 235U fragments just shared neutrons?

– Both nuclei are neutron rich, Z/A ~ 0.39• Stable nuclei in this region prefer Z/A ~0.41

– Fragments tend to shed excess neutrons at the instant of fission

– This leads to the emission of prompt neutrons:

– Unstable fragments can lead to the emission of delayed neutrons, following decay

• About one per 100 fissions

CsRbnU 14195235

nucleus <# prompt neutrons>233U 2.48235U 2.42

239Pu 2.86

Distribution is approximately Gaussian, consistent with an evaporation process.

5

Controlled Fission

• The neutrons produced in a fission reaction are fast (few MeV)

• If we can moderate or slow down the neutrons, then they can initiate additional fission reactions– Slower neutrons have higher capture cross sections

– This is the idea behind a controlled chain reaction used in nuclear production

• E. Fermi (1942)

– Early reactors used carbon as a moderator • Light nucleus, large energy transfer in collision

• Interleaved U and C blocks formed a pile (238U capture resonances)

• Neutron Reproduction Factor k:– The change in the number of thermal neutrons from one

generation of reactions to the next04/19/23 Physics 590B - Fall 2014

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Controlled Fission Cyclek<1 subcriticalk=1 criticalk>1 supercritical

Start

=1.88 for enriched U235U/238U ~ 3%

235U/238U ~ 0.72%

~1.03

some neutrons will induce fission in 238U

thermal neutrons induce fission in 235U

238U resonances in 10-100eV region

captures on moderator

p~0.9

f~0.9

))(( tf llpfk

pfk

11

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Critical Size

• Minimizing the surface area of the pile will minimize neutron leakage

• Leakage depends on how far a neutron can travel without being absorbed (called the migration length M)

– For a graphite pile Ls=18.7cm, Ld=50.8cm

• For a spherical pile can guess

• There will be a critical size corresponding to k=1

• For a spherical arrangement this is about RC=5m

21

22sd LLM

Slow down fast neutrons

Slow neutron diffusion

2

2

R

Mkk

1

k

MRC

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Timescales and Control

• The neutrons are characterized by a time constant that involves both moderation (10-6 s) and absorption (10-3 s)

• If you have N neutrons at t=0, you have kN at t=

• So if k>1 the number of neutrons will grow exponentially with a timescale of order ms….

• Solution is to use Cd control rods to absorb neutrons– Reactor is subcritical for prompt neutrons

– Delayed neutrons (with longer timescale) make it critical

/)()( tkeNtN

dtNkNdN

10

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9

A Natural Fission Reactor

• The natural abundance of 235U relative to 238U is about 0.72%– Moon rocks show same abundance

• Oklo mine (Gabon, Africa) shows an unusually low abundance of 235U (3 below the mean), some places as low as 0.44%!– No known chemical process should change the natural ratio like this

• About 2x109 years ago, the natural abundance of 235U relative to 238U was about 3%– A “natural” fission reactor could have operated, using groundwater as

a moderator

– Low power (0.01MW) or it would boil away the groundwater

– Burned for about 106 years

– Consumed about 5 tons of 235U04/19/23 Physics 590B - Fall 2014

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Oklo Mine - Isotopic Abundances

• Isotopic abundances of fission fragments consistent with 235U fission!

• Now that’s “green energy”!

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Fission Reactor Technology (I)

• Classify fission reactors by MODERATOR– Graphite moderated

• Older design, safety issues (Chernobyl)

– Heavy water (D20) moderated

• Can use unenriched Uranium– Bypass international restrictions on enriching Uranium

– Produce more Pu as a byproduct

– Light water moderated• Require enriched Uranium

• Negative feedback stabilizes reactor– Density of water falls as temperature increases

– Molten Salt Reactors (Li, Be)• Very compact design (aircraft)

• Simple design, low pressure

– Liquid Metal (fast reactors, unmoderated)• Soviet nuclear submarines (Alfa class)04/19/23 Physics 590B - Fall 2014

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Fission Reactor Technology (II)

• Classify fission reactors by COOLING:– Pressurized Water

• Coolant kept under pressure to keep it from boiling

– Three Mile Island was this type

– Boiling Water• Coolant is allowed to boil, steam

pressure used to regulate

– Pool Type

– Liquid Metal• Fast reactors (no moderator)

• Na, Pb, Pb-Bi, etc

– Gas Cooled (He, N, CO2,…)

– Molten Salt (LiF, BeF2)

• Fuel dissolved in coolant04/19/23 Physics 590B - Fall 2014

13

Nuclear Fuel

• Uranium and Plutonium are used in a variety of forms as a nuclear fuel– Uranium Oxide

• Enrichment varies

– MOX Fuel• Mixture of Pu and depleted U

• Alternative to LEU for LWR’s

• Used by England, France andRussia, India, Japan to a lesser extent

• China plans fast breeders withreprocessing

– Molten Salts

– TRISO• Pebble-bed reactors

UOX

C

SiC

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Uranium Enrichment

• Mined Uranium ore is refined and converted to UF6

– USA, France, UK, Russia, Iran(?)

– Highly dangerous and corrosive, shipped as a solid crystal

– UF6 gas can be 235U enriched by diffusion or centrifuge

– Back to UO2 (pellets)

64283 UFFUFHFUOOU

238U

235U

Zippe centrifuge04/19/23 Physics 590B - Fall 2014

15

Breeder Reactors

“Breeder” fission reactors can essentially create their own fuel from an initial fuel charge and 238U.

Refueling involves reprocessing and adding a new charge of 238U

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Nuclear Waste

• 99% of the radioactive waste is in the nuclear fuel rods

• Stored locally in pools of boric acid– Can’t store too much together or

they might go critical!

• Overcrowding of pools has led to “dry cask” storage– Rods moved after ~5yrs in cooling

pond

– Also stored on site

• Currently no permanent solution to waste storage

Dry Cask Storage

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Yucca MountainGeologically stable for ~10k years (expected)

Underground storage facility constructed by tunnel boring into the mountain.

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Dumped Reactors (USSR)

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Chernobyl

• 26 April 1986:

• Explosion and fire in reactor #4 at Chernobyl nuclear facility near Pripyat, Ukraine

• 400 times more fallout that Hiroshima

• Catastrophic power excursion caused steam explosion

• Ironically caused by a failed safety test prior to shutdown for refueling

steam separator

steam

coolingwater

control rods

radiation shieldand containment building

RMB-1000 Nuclear Reactor

04/19/23 Physics 590B - Fall 2014

Modern Reactor Designs

• Passive Protection – designed with negative feedback to keep system stable– No diesel generators required in event of power failure

– Can operate for long period of time without human intervention

– Passive circulation relies on gravity (not pumps)

20

Westinghouse AP1000 approved by NRC in 2005. Units in China already under constrcution, planned operation in 2013-15.

Fourteen applications for operating licenses pending in US. (Georgia plant loan guarantees)04/19/23 Physics 590B - Fall 2014

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Fission Explosives

• Nuclear weapons require much more highly enriched Uranium– Need energy release from a supercritical mass faster than the mass

is blown apart

– A crude device could be built with ~20% 235U, “modern” weapons use >85%

• Collect subcritical pieces into a critical assembly

Frank Spedding Harley Wilhelm

~2M lbs of pure Uranium 1942-4504/19/23 Physics 590B - Fall 2014

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Fission Weapon Designs

Pu coreNeutron Initiator

“Slow” Explosive

“Fast” Explosive

Tamper/Pusher

Shockwave

“Fat Man” (Nagasaki)

“Little Boy” (Hiroshima)

nCBePo 129)(

~20kT TNT ea., ~1kg of material consumed

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Fusion

• Instead of splitting large nuclei, what if we combine light elements

• Fusion has many keyadvantages overfission:– Light nuclei (p,d,t)

easy to obtain

– End products light andstable

• However, in orderto get nuclei to fuseyou have to overcomethe Coulomb barrier between them.

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Coulomb Repulsion (Again!)

• Consider:– Using the “two spheres touching” model:

– So if we collide a beam of 20Ne on a 20Ne target at 21.2 MeV we would get back

– Almost double our investment!

– Why doesn’t this work as a power source?• Doesn’t take inefficiencies into account

• High intensity beams difficult to produce

• At best you could get a few Watts…

CaNeNe 402020 (Q = 20.7 MeV)

MeVfm

fmMeVR

eZZV

fmRRR

22186

10441

4

186202512

2221

0

2131

..

)().(

.)(.

MeVMeVMeVTQE 941221720 ...

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Thermonuclear Fusion

• What if you were to heat a container of 20Ne to an average kinetic energy of 21.2MeV– Obtain a much higher particle density!

– We want:

– As a rule of thumb, at room temp. kBT ~ 1/40 eV

– For 20Ne, TF ~ 1011K

• Core temperature of our sun ~107K

– This will be difficult!• Still, if you want to compete with commercial fission reactors at

~1GW, this is what you have to do

• Must be a good idea, the stars do it…

CaNeNe 402020

MeVTk

MeVTk

FB

FB

172

1221

2

3

.

.

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Basic Fusion Processes (I)

• The most basic fusion process we can think of is:

– Not possible! 2He is unstable…

• A possible reaction is:

– Requires the weak interaction to come into play

– This reaction will be rate limiting!

• Also possible:

– 4He excited state high in energy, so photon necessary for energy balance

– Q is greater than the n,p separation energy for 4He

– This reaction is unlikely

Hepp 2

eeHpp 2

HeHH 422(Q=23.8 MeV)

eeHpp 2(Q=1.44 MeV)

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Basic Fusion Processes (II)

• More likely deuterium reactions are:

• d-t reactions are also possible:

– Large energy release, good candidate for energy production

– Have to overcome barrier of:

– Don’t need to overcome this, just come close so the tunneling probability is appreciable.

pHHH

nHeHH

322

322(Q=3.3 MeV)

(Q=4.0 MeV)

nHeHH 432(Q=17.6 MeV)

MeVfm

fmMeVR

eZZV 4260

32251

1441

4

131

31

2221

0

.)(.

)().(

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Kinematics

• If the initial kinetic energy is low compared to the Q value, so we can write:

• For the d-t reaction, <En> ~ 14.1MeV– This energy can be difficult to extract

n

HeHeHe

He

nnn

HeHenn

HeHenn

mmQ

vm

mmQ

vm

vmvm

Qvmvm

12

1

12

1

2

1

2

1

2

2

22

The lightest particle will carry away most of the Q-value!

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Fusion Cross Sections

• For particles interacting at thermal energies, the reaction will most likely occur away from any resonances

• The basic fusion cross section can be written as:

– Where G is the same Gamow factor we encountered in decay

– For Ek << VB we can approximate:

– The proportionality factor will account for statistical factors, spins, etc.

Gev

22

1

hv

ZZeG Xa

0

2

4

v = relative velocity

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Fusion Rates (I)• The rate a reaction will proceed at depends on the cross

section:

– Just rewriting what we had before…

• For a thermal collection of matter, the velocities will be distributed according to a Maxwell-Boltzmann factor:

• So the relevant quantity is the thermal averaged cross section:

vnINR 0

No. of target atoms

Intensity (s-1 cm-2)density (cm-3)

rel. velocity (cm s-1)

Tkmv

Bevn 22

)(

0 0

2222

1dEeedvvee

vv Tk

EGTk

mvG BB

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Fusion Rates (II)

• The fusion rate for a process will depend on the interplay between the cross section and the Maxwell-Boltzmann distribution– MB peaked low

– v grows for higher energies

0

2221

dvveev

v Tkmv

G B

(For asymmetric systems we need two Boltzmann distributions and as a function of vrel)04/19/23 Physics 590B - Fall 2014

32

Fusion Rates (III)107 K 108 K

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Solar Fusion (I)

• We could learn a lot from the sun…

• The proton-proton cycle is the primary process by which the sun produces energy:

– Bottleneck for the whole process:

– At T~1.5x107K in the core of the sun, kBT~1keV

– The Boltzmann tail helps you reach higher energies where the cross section is larger

– Reaction rate about 1038/s in the sun

• Next step:

– 2H+2H unlikely at this point, concentration of 2H too low• D/H < 10-5

eeHpp 2(Q=1.44 MeV)

MeV) (1 10 tokeV) (1 10~ 2333 bb

HeHp 32 (Q=5.49 MeV)

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Solar Fusion (II)

• Next step:

• Instead:

• The net result of these interactions is:

• Other reactions are possible:

HepLiHep 343 (4Li not stable – no help!)

pHeHeHe 2433(Q=12.86 MeV)

eeHep 224 4 (Q=26.7 MeV)

BeHeHe 743

HeBe

eBeB

BpBe

e

48

88

87

2

HepLi

vLieBe e

47

77

2

(same final state, same Q value)

monoenergetic neutrinos!

eeHeHep 43 (very rare, “hep” neutrinos)

(pp I)

(pp III)(pp II)

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The C-N-O Cycle• The presence of 12C in the stellar interior can act as a

catalyst to fusion:

• All of these reactions produce neutrinos, which immediately escape the sun…

HeCpN

eNO

OpN

NpC

eCN

NpC

e

e

41215

1515

1514

1413

1313

1312

eeHep 224 4

No deuterium bottleneck!

However, the Coulomb barrier is 6-7 times higher

This process dominates at higher T.

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Solar Neutrinos

Ray Davies (1964) – deficit of neutrinos from inverse beta process AreCl 3737 ),(Kamiokande, Gallex, SAGE, etc (80’s-90’s): confirm deficit

SNO (2001) – not all neutrinos are electron neutrinos when they reach Earth!!

Kamland – verified neutrino oscillations theory.04/19/23 Physics 590B - Fall 2014

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Neutrino Oscillations

• The neutrino flavor eigenstates are not the same as the mass eigenstates:

• Neutrinos born as e can be detected as

(E=1GeV, m2=0.005 eV2)

321

,,

,,*

iU

eU

ii

iii

)0()(

)0()(2/

)(

2

iELim

i

ixpEti

i

ieL

et

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Fusion PowerMagnetic Confinement:

“Mirror”Tokamak

Atmos. Formation Compression Ignition Burn

Inertial Confinement:

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Blanket

Magnet System

Vacuum Vessel

Person

R=6.2 mIp=15 MA

Pfus=500 MW

Divertor

30 m

24 mITER

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Thermonuclear Weapons

• Essentially a daisy-chain of a fission and fusion bomb:

fissionbomb

fusion fuel:238U, LiD, 235U

primary fires

X-rays reflect into LiD fission fuel casing (polystytrene)

Plasma ignites 235U sparkplug

Li converted to 3H, fusion begins, tamper fissions

nHHenLi 346

04/19/23 Physics 590B - Fall 2014

BACKUP

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42

Fission Lifetimes

• If it is energetically favorable, why don’t nuclei spontaneously fall apart?– For 238U, t1/2 = 4.5 x 109 years for decay, but 1016 years for

fission!

• The Coulomb barrier inhibits fission in much the same way as for decay– Barrier height for 238U decay to 119Pd estimate:

– The 214 MeV energy release makes many final states available, however the barrier height makes tunneling unlikely!

MeVfm

fmMeVR

eZZV

fmRRR

250212

46441

4

12121192512

2221

0

2131

.

)().(

.)(.

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Spontaneous vs. Induced Fission

• Classify fission processes according to the barrier height:

• Spontaneous Fission– If E ~ Coulomb, fission will compete with other decay processes.

This is not observed for naturally occurring nuclei, but becomes important around A~300

• Induced Fission– If E < Coulomb, fission can be induced by the absorption of a

neutron or gamma ray

– Activation Energy is the height of the fission barrier above the ground state

decay alphamostly 1028

105415

9

23892

21

21

ySFt

ytU

.)(

.:

decay alpha 80% 5

1

21

21261

107

msSFt

mstBh

)(:

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Activation Energy (I)

• The liquid-drop model can predict the average behavior– Of course, shell effects will modify this

– Get a quantitative feel for the fission process

– Stretch a nucleus, keeping the volume constant

– As the nucleus is stretched, the surface area changes:

– The dominant change in the binding energy comes from the surface area and Coulomb terms:

2

3

4abV

a

b

21

11 )()( RbRa

22

5

214 RS

23

12

5

1

5

20 3

2

AZaAaBBE CS)()(

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Activation Energy (II)

• If the second term dominates the first term, we gain energy and the nucleus will be unstable

23

12

5

1

5

20 3

2

AZaAaBBE CS)()(

47

5

1

5

2

2

3

123

2

A

Z

AZaAa CS

spontaneous fission condition

04/19/23 Physics 590B - Fall 2014