nuclear fission

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

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Nuclear Fission

1/v

235U thermal cross sectionsfission 584 b.scattering 9 b.radiative capture 97 b.

Fast neutrons should be moderated.

Fission Barriers

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Nuclear Fission

• Q for 235U + n 236U is 6.54478 MeV.• Table 13.1 in Krane: Activation energy EA for 236U 6.2 MeV (Liquid drop + shell) 235U can be fissioned with zero-energy neutrons.

• Q for 238U + n 239U is 4.??? MeV.• EA for 239U 6.6 MeV MeV neutrons are needed.• Pairing term: = ??? (Fig. 13.11 in Krane).• What about 232Pa and 231Pa? (odd Z).• Odd-N nuclei have in general much larger thermal neutron cross sections than even-N nuclei (Table 13.1 in Krane).

Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).


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Nuclear Fission

f,Th 584 2.7x10-6 700 0.019 b

Why not use it?Why not use it?


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Nuclear Fission• 235U + n 93Rb + 141Cs + 2n• Q = ????• What if other fragments?• Different number of neutrons.• Take 200 MeV as an average.

66 MeV 98 MeV

miscalibrated

Heavyfragments

Lightfragments


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Nuclear Fission• Mean neutron energy 2 MeV.• 2.4 neutrons per fission (average) 5 MeV average kinetic energy carried by prompt neutrons per fission.

• Show that the average momentum carried by a neutron is only 1.5 % that carried by a fragment. • Thus neglecting neutron momenta, show that the ratio between kinetic energies of the two fragments is the inverse of the ratio of their masses.

1

2

2

1

mm

EE

14095

9866

HW 45HW 45


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Nuclear Fission

Distribution of fission energy

Krane sums

them up as

decays.Lost … !

Enge

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Segrè

Lost … !

Nuclear Fission

• How much is recoverable?How much is recoverable?• What about capture gammas? What about capture gammas? (produced by (produced by -1 neutrons)-1 neutrons)• Why c < (a+b) ?Why c < (a+b) ?

Distribution of fission energy

abc


8Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

• Recoverable energy release 200 MeV per 235U fission.• Fission rate = 2.7x1021 P fissions per day. P in MW.

• Burnup rateBurnup rate: 1.05 P g/day. P in MW.

• Capture-to-fission ratio:

• Consumption rateConsumption rate: 1.05(1+) P g/day.• 1000 MW reactor.• 3.1x1019 fissions per second, or 0.012 gram of 235U per second.• Two neutrinos are expected immediately from the decay of the two fission products, what is the minimum flux of neutrinos expected at 1 km from the reactor.

4.8x1012 m-2s-1

Nuclear Fission

)()(

)(EE

Ef

HW 46HW 46

9Nuclear Reactors, BAU, 1st Semester, 2007-2008 (Saed Dababneh).

Nuclear Fission

• 3.1x1010 fissions per second per W.• In thermal reactor, majority of fissions occur in thermal energy region, and are maximum.• Total fission rate in a thermal reactor of volume V

• Thermal reactor powerThermal reactor power (quick calculation) (quick calculation)

fV

10101.3 xV

P fth


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Controlled Fission• 235U + n X + Y + (~2.4)n• Moderation of second generation neutrons Chain reaction.• Net change in number of neutrons from one generation to the next k (neutron reproduction factor).

• k 1 Chain reaction.• Water, D2O or graphite moderator.• k < 1 subcritical system.• k = 1 critical system.• k > 1 supercritical system.• For steady release of energy (steady-state operation) we need k =1.

Fast second generation neutrons

Infinite medium (ignoring leakage at the surface).

Chain reacting pileChain reacting pile


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Probability for a thermal neutron to cause fission on 235U is

Controlled Fission235U thermal cross sectionsfission 584 b.scattering 9 b.radiative capture 97 b.

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f

f

If each fission produces an average of neutrons, then the mean number of fission neutrons produced per thermal neutron =

1f

f

a

f <

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Controlled Fission235U

238U

• Assume Assume natural natural uranium:uranium:99.2745% 238U, 0.7200% 235U.

• f / N = (0.992745)(0) + (0.0072)(584)

= 4.20 b.• / N = (0.992745)(2.75) +

(0.0072)(97) = 3.43 b.

Thermal f = 0 b 584 bThermal = 2.75 b 97 b


N

NN

yyxx

yyxxyx

)(

24 R

24 R

Using the experimental elastic scattering data the radius of the nucleus can be estimated.

Doppler effect?Doppler effect?

Moderation (to compare x-section)

1H

(n,) (n,)

(n,n)(n,n)2H

• Resonances?


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f

f• Probability for a thermal neutron to cause fission

• For natural uraniumnatural uranium

• If each fission produces an average of = 2.4 neutrons, then the mean number of fission neutrons produced per thermal neutron = = 2.4 x 0.55 1.3

• This is close to 1. If neutrons are still to be lost, there is a danger of losing criticality.• For enriched uraniumenriched uranium (235U = 3%) = ????? (> 1.3).• In this case is further from 1 and allowing for more neutrons to be lost while maintaining criticality.

55.043.320.4

20.4

Controlled Fission

f

f


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Controlled Fission• N thermalthermal neutrons in one generation have produced so far have produced so far NN fast neutrons.fast neutrons.• Some of these fast neutrons can cause 238U fission more fast neutrons fast fission factor = (= 1.03 for natural uranium).• Now we have Now we have NN fast neutrons. fast neutrons.• We need to moderate these fast neutrons use graphite for 2 MeV neutrons we need ??? collisions. How many for 1 MeV neutrons?• The neutron will pass through the 10 - 100 eV region during the moderation process. This energy region has many strong 238U capture resonances (up to 1000 b) Can not mix uranium and graphite as powders.• In graphite, an average distance of 19 cm is needed for thermalization the resonance escape probability p ( 0.9).


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Controlled Fission• Now we have Now we have ppN N thermal neutrons.thermal neutrons.• Graphite must not be too large to capture thermal neutrons; when thermalized, neutrons should have reached the fuel.• Graphite thermal cross section = 0.0034 b, but there is a lot of it present.• Capture can also occur in the material encapsulating the fuel elements.• The thermal utilization factor f ( 0.9) gives the fraction of thermal neutrons that are actually available for the fuel.• Now we have Now we have fpfpNN thermal neutrons thermal neutrons, could be > or < N thus determining the criticality of the reactor.

k = fp The four-factor formula.The four-factor formula.

k = fp(1-lfast)(1-lthermal)Fractions lost at surface


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x 1.03x 1.03Fast fission Fast fission factor “factor “””

x 0.9x 0.9Resonance Resonance

escape escape probability ”p”probability ”p”

x 0.9x 0.9Thermal Thermal

utilization utilization factor “f”factor “f”

x

What is:• Migration length?• Critical size?How does the geometry affect the reproduction factor?

Neutron reproduction

factork = 1.000


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Controlled FissionTime scale for neutron multiplicationTime scale for neutron multiplication• Time includes moderation time (~10-6 s) and diffusion time of thermal neutrons (~10-3 s).

Time Average number of thermal neutrons t Nt + kNt + 2 k2N

• For a short time dt

• Show thatShow that

NkN

dtdN

tkeNtN )1(0)(


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• k = 1 N is constant (Desired).• k < 1 N decays exponentially.• k > 1 N grows exponentially with time constant / (k-1).• k = 1.01 (slightly supercritical) e(0.01/0.001)t = e10 = 22026 in 1s. in 1s. • Cd is highly absorptive of thermal neutrons.• Design the reactor to be slightly subcritical for prompt neutrons.• The “few” “delayed” neutrons will be used to achieve criticality, allowing enough time tomanipulate the controlrods.

tkeNtN )1(0)(

Controlled Fission

Dang

erou

s

Dang

erou

s

Cd control rodsCd control rods


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Fission ReactorsEssential elements:Essential elements:• Fuel (fissile material).• Moderator (not in reactors using fast neutrons).• Reflector (to reduce leakage and critical size).• Containment vessel (to prevent leakage of waste).• Shielding (for neutrons and ’s).• Coolant.• Control system.• Emergency systems (to prevent runaway during failure).

Core


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Fission ReactorsTypes of reactors:Types of reactors:Used for what?Used for what?• Power reactors: extract kinetic energy of fragments as heat boil water steam drives turbine electricity.• Research reactors: low power (1-10 MW) to generate neutrons (~1013 n.cm-2.s-1 or higher) for research.• Converters: Convert non-thermally-fissionable material to a thermally-fissionable material.

_2393.2

_239min23239238

Pu

NpUnU

dFertile

f,th = 742 b


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Fission Reactors

_23327

_233min22233232

U

PaThnTh

dFertile

• If = 2 Conversion and fission.• If > 2 Breeder reactor.• 239Pu: Thermal neutrons ( = 2.1) hard for breeding.

Fast neutrons ( = 3) possible breeding fast breeder reactors.

f,th = 530 b

After sufficient time of breeding, fissile material can be easily (chemically) separated from fertile material.Compare to separating 235U from 238U.


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Fission ReactorsWhat neutron energy?What neutron energy?• Thermal, intermediate (eV – keV), fast reactors.• Large, smaller, smaller but more fuel.What fuel?What fuel?• Natural uranium, enriched uranium, 233U, 239Pu.

From converter or breeder reactor.How???

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Fission ReactorsWhat moderator?What moderator?1. Cheap and abundant.2. Chemically stable.3. Very low mass (~1).4. High density.5. Minimal neutron capture cross section.• Graphite (1,2,4,5) increase amount to compensate 3.• Water (1,2,3,4) but n + p d + enriched uranium.• D2O (heavy water) has low capture cross section

natural uranium, but if capture occurs, produces tritium.

• Be and BeO, but poisonous.


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Fission ReactorsWhat assembly?What assembly?• Heterogeneous: moderator and fuel are lumped. • Homogeneous: moderator and fuel are mixed together.• In homogeneous systems, it is easier to calculate p and

f for example, but a homogeneous natural uranium-graphite mixture can not go critical.

What coolant?What coolant?• Coolant prevents meltdown of the core.• It transfers heat in power reactors.• Why pressurized-water reactors.• Why liquid sodium?


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Boiling water reactorBoiling water reactor

Pressurized Pressurized water water

reactorreactor

• Light water reactors.Light water reactors.• Both use “light” water as Both use “light” water as coolant and as moderator, coolant and as moderator, thus enriched (2-3%)thus enriched (2-3%) uranium is used.uranium is used.• Common in the US.Common in the US.


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CANDU CANDU reactorreactor

Gas Gas cooled cooled reactorreactor

• Canada has DCanada has D22O O and natural uranium.and natural uranium.

• Most Most power power reactors in reactors in GB are GB are graphite graphite moderated moderated gas-gas-cooled.cooled.


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Breeder Breeder reactorreactor

• Liquid sodium cooled, fast breeder reactor.Liquid sodium cooled, fast breeder reactor.• Blanket contains the fertile Blanket contains the fertile 238238UU..• Water should not mix with sodium.Water should not mix with sodium.

nuclear fission

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