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
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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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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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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
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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
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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
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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)
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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
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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
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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
