Transcript Slide 1

While for large r, after the fragments have been scissioned

V(r)
r
for small r
r
r
Z1Z 2e
V (r ) 
r
for large r
separation r
2
For such quadrupole
distortions the figure
shows the energy of
deformation (as a factor
of the original sphere’s
surface energy Es)
plotted against 
for different values of
the fission parameter x.
When x > 1
(Z2/A>49)
the nuclei are
completely unstable
to such distortions.
Z2/A=36
such unstable states
decay in characteristic
nuclear times ~10-22 sec
Z2/A=49
Tunneling does allow spontaneous
fission, but it must compete with
other decay mechanisms (-decay)
The potential energy V(r) = constant-B
as a function of the separation, r, between fragments.
Thermal neutrons
E< 1 eV
Slow neutrons
E ~ 1 keV
Fast neutrons
E ~ 100 keV – 10 MeV
“Thermal neutrons”
(slowed by interactions
with any material they
pass through) have been
demonstrated to be
particularly effective.
Cross section 
The incident neutron itself need not be of high energy.
Typical
of decay
Products
& nuclear
reactions
incident particle velocity, v
This merely reflects the general ~1/v behavior
we have noted for all cross sections!
At such low excitation there may be barely enough available
energy to drive the two fragments of the nucleus apart.
Division can only proceed
if as much binding energy as possible
is transformed into the kinetic energy separating them out.
(so MOST of the available Q goes into the kinetic energy of the fragments!)
Thus the individual nucleons
settle into the lowest possible energy configurations
involving the most tightly bound final states.
There is a strong tendency to produce a heavy fragment of
A ~ 140 (with double magic numbers N = 82 and Z = 50).
A possible (and observed) spontaneous fission reaction
U 2 46 Pd
238
119
92
8.5 MeV/A
7.5 MeV/A
Gains ~1 MeV per nucleon!
2119 MeV = 238 MeV
released by splitting
119Pd
238U
238 MeV represented an estimate of the maximum available energy
for symmetric fission.
For the observed
distribution
of final states
the typical average is
~200 MeV per fission.
This 200 MeV is distributed approximately as:
Fragment kinetic energy
Prompt neutrons
Prompt gamma rays
Radioactive decay fragments
165 MeV
5 MeV
7 MeV
25 MeV
235U
Isobars off the valley of stability
(dark squares on preceding slide)
b-decay to a more stable state.
 and b decays can leave a daughter in an excited nuclear state
1/2187W
2-
b-
b-
198Au
0.68610
0.61890
b-
1.088 MeV
b-
0.20625
0.412 MeV
0.13425
5/2+
187Re
0+
198Hg
n+ 92 U  92 U * 56 Ba + 36 Kr + 3n
235
236
143
90
With the fission fragments radioactive, a decay sequence to stable nuclei must follow
Ba

Kr

143
56
90
36
La + e + 
143
Ce + e + 
58
143
Pr + e + 
59
143
Nd + e + 
59
90
Rb + e + 
37
90
Sr + e + 
38
90
Y +e +
39
90
Zr + e + 
40
143
-
57
n+ 92 U  92 U * 60 Nd + 40 Zr + 8e + 8 + 3n
235
236
143
90
n+ 92 U  92 U * 55 Cs+37 Rb + 2n
235
236
141
93
With the fission fragments radioactive, a decay sequence to stable nuclei must follow
141
55
Cs

b,
25 sec
0.03%
141
56
Ba
Cs + n 
b,
65 sec
140
55
93
37
Rb

b,
18 min

b,
6 sec
1.40%
92
37
93
38
Sr

7 min
b,
Rb + n 
b,
5 sec
57
La
13 d
Ba
93
Y
39

b,
92
38

b,
140
b,
140
56

141
4 hr
141
Sr
10 hr

3 hr
b,
La
40 h
93
40

b,
140
Ce
58
57

b,
141
33 d
Zr
92
Y
39

b,
106 yr

b,
4h
59
58
Pr
Ce
93
Nb
41
92
40
Zr
n+ 92 U  92 U * 58 Ce+ 59 Pr + 8e + 8 + 8 + 2n
sometimes + 3n or + 4n
235
236
140
141
For 235U fission, average number of prompt neutrons ~ 2.5
n+ 92 U  92 U * 56 Cs+ 36 Kr + 3n
235
236
143
90
n+ 92 U  92 U * 55 Cs+37 Rb + 2n
235
236
141
93
n+ 92 U  92 U * 57 La + 35 Br + 2n
235
236
139
95
n+ 92 U  92 U * 54 Xe+38 Sr + 2n
235
236
139
95
with a small number of additional delayed neutrons.
with every neutron freed comes the possibility of additional fission events
This avalanche is the chain reaction.
235U
will fission (n,f)
at all energies of the absorbed neutron.
It is a FISSILE material.
However such a reaction cannot occur in
natural uranium (0.7% 235U, 99.3% 238U)
Total (t) and fission (f) cross sections of 235U.
1 b = 10-24 cm2
Notice:
238U
has a threshold for fission (n,f) at a neutron energy of 1MeV.
The difference between these two isotopes of uranium
is explained by the presence of the pairing term
in the semi-empirical mass formula.
  + a pair A
-3 / 4
for Z even, N even
 -a pair A
for Z odd, N odd
0
for A odd
-3 / 4
Like nucleons couple pairwise into especially stable configurations.
Note the strong
resonant capture of
neutrons (n, ) in
the energy range
10-100 eV
(particularly
for 238U
where the
cross-section
reaches
high values)
The fission neutron energy spectrum peaks at around 1 MeV
At 1 MeV
the inelastic
cross-section
(n,n') in 238U
exceeds the
fission
cross-section.
This effectively
prevents fission
from occurring
in 238U.
only
the
Natural uranium (0.7% 235U, 99.3% 238U)
undergoes thermal fission
Fission produces mostly fast neutrons
Mev
but is most efficiently
induced by slow neutrons
E (eV)
Consider fission neutrons created deep enough
in a lump of natural uranium
that we’ll just (for now) ignore that
some neutrons may simply escaping from the sample.
100
10
1
Cross-section (barns)
1000
The processes competing with
neutron-induced fusion
have approximate cross-sections
(read from the graphs at right) of
235U
(n,n) elastic scattering ~ 5 barn
(n,n’) inelastic scattering ~ 3 barn
(n,)
~0.2barn
(n,f) fission
~ 2 barn
238U
(n,n) elastic scattering ~ 5 barn
(n,n’) inelastic scattering ~ 2 barn
(n,)
~0.2barn
(n,f) fission
~0.6barn
1000
100
10
1
0.1
Cross-section (barns)
10000
Giving a relative probability to each of:
235U
238U
(n,n) elastic scattering
(n,n’) inelastic scattering
(n,)
(n,f) fission
6.7
1.7
0.3
3.3
(n,n) elastic scattering
(n,n’) inelastic scattering
(n,)
(n,f) fission
8.3
3.3
0.3
1
 0.7/99.3
Of the first 100 fission neutrons we start with
~98 are captured in the dominant 238U
238U
235U
(n,n) elastic scattering
(n,n’) inelastic scattering
(n,)
(n,f) fission
63
25
2
8
(n,n) elastic scattering
(n,n’) inelastic scattering
(n,)
(n,f) fission
1
0
0
0
With 2-3 neutrons generated by each fission,
only ~20 neutrons in the second generation
- this is insufficient to sustain a chain reaction.
only 8 of
these
captures
result
in fission
FAST REACTOR
Enriching the 235U content
a 50-50 mix of the two isotopes will sustain a chain reaction
(most fission events occurring now in 235U
by neutron energies in the range 0.3 - 2.0 keV.
THERMAL REACTOR moderating the neutrons to thermal speeds
mixing natural uranium with a material to slow
(but not absorb) neutrons to lower energies
where the fission cross-section for 235U is large.
Most fissions are then induced by neutrons
with thermal energies (~0.025 eV).
Granulated powders can be mixed for this purpose.
Powdered uranium
Or blocks of uranium fuel can be alternately stacked
with graphite to form a nuclear pile.
FUEL
Moderator
(Graphite)
FUEL
1. Starting with  neutrons/fission
2. Avg of  neutrons after fast fission
3. p survive thermalization
4. pf number captured in 235U
238U
Moderator
(Graphite)
FUEL
5. k = pf(f /total)
number producing fission
235U
One fission event produces
k = pf(f /total)
secondary fission events.
k is the reproduction factor.
A chain reaction requires k1.
If k=1 the core is “critical” and self-sustaining.
Typical values for natural uranium/graphite piles are
  2.47
f  0.88
  1.02 p  0.89
 f /  t  0.54
k=1.07
Uranium is not dumped into the core like coal
shoveled into a furnace. Instead it is processed
and formed into fuel pellets (~pencil eraser size).
The fuel pellets are stacked inside hollow metal
tubes to form fuel rods 11 to 25 feet in length.
Before it is used in the reactor, the uranium fuel is not very radioactive.
The fuel rods are arranged in a regular lattice inside the moderator.
The rods are typically 2-3 cm in diameter and
spaced about 25 cm apart.
The rods metal sheath or cladding –
most commonly stainless steel or alloys of zirconium.
This cladding supports the fuel mechanically,
prevents release of radioactive fission products into the coolant stream and
provides extended surface contact
with the coolant in order to promote effective heat transfer.
A single fuel rod cannot generate
enough heat to make the amount of
electricity needed from a power
plant. Fuel rods are carefully
bound together in assemblies,
each of which can contain over
200 fuel rods. The assemblies hold the
fuel rods apart so that when
they are submerged in the reactor core,
water can flow between them.
In nuclear power plants, the moderator is often water
(though some types do still use graphite).
Fuel cell channels
in face
of reactor core.
Control rods
slide in or out between the fuel rods
to regulate the chain reaction.
contain cadmium or boron (high cross section
for neutron absorption, without fission).
e.g., natural Boron is 20% 10B
with a cross section
for thermal neutrons of
3840 b for the process
10
B + n Li * +
7
Control rods act like sponges to absorb excess neutrons.
When the core temperature drops too low,
the control rods are slowly pulled out of the core,
and fewer neutrons are absorbed.
When the temperature in the core rises,
the rods are slowly inserted.
To maintain a controlled nuclear chain reaction,
the control rods are manipulated until each fission
results in just one neutron on average, all other
neutrons effectively absorbed by the control rods.
Temperature changes in the core are generally very gradual.
However should monitors detect a sudden change in temperature,
the reactor immediately shuts down automatically by dropping
all the control rods into the core. A shutdown takes only
seconds and halts the nuclear chain reaction.
This very common type makes use
of the excellent properties of water as
both coolant and moderator (ordinary
water does absorb neutrons –
converting hydrogen into deuterium).
The Boiling Water Reactor (BWR)
allows the water to boil in the reactor
core and uses the steam to drive the
turbines.
The highest temperature possible for liquid water (critical temperature
374°C) is a limitation for devices that use water to convey heat.
The core must be contained within a pressure vessel of welded steel
(typically withstanding pressures of about 1.55 107 Pa or 153 bar.
Furthermore recall: the Carnot engine efficiency  is
  1 - T2 / T1
In this ideal case the heat is received isothermally (the working fluid at T1)
but rejected isothermally (at T2) with all processes reversible.
No real power plant operates on an ideal Carnot cycle, but the expression
shows the higher T1, the higher the efficiency (T2 cannot be lower than
the outside temperature).
1st land based pressurized water reactor: Shippingport USA (1957).
Pressure vessels are enormous
with 9 inch thick walls, often
weighing more than 300 tons.
The pressure vessel surrounds
and protects the reactor core,
providing a safety barrier and
holding the fuel assemblies,
control rods, and coolant.
Pressure vessels are made of carbon steel and lined with a layer of
stainless steel to prevent rust. The pressure vessel is located inside
the containment building, a thick concrete structure reinforced with
steel bars.
A Fast Reactor has no moderator
and consequently a much smaller core.
The very high power involved means that
liquid metals have to be used as coolants!
Liquid sodium is the most common but
has the disadvantage of becoming
radioactive itself through
23Na(n, )24Na.
As well as generating power
fast reactors are used for
breeding fissile material.
If uranium fission reactors used as sole source of
electrical power needs
all high-grade ores used up within a few decades!
Breeder Reactors
Fermi, Zinn (1944)
Can fissile nuclei be grown? (the result of any nuclear reaction)
Can we create fissile material as a by product of any reaction?
The parent nuclei that spawns the fissile material
is described as being a fertile nuclide.
Example: build a reactor core that runs on 239Pu (the fuel)
packed within a bed of 238U (the fertile nuclide)
• =2.91 fast neutrons/239PU fission
Only one of these on average producing an additional fission
is sufficient for sustainability.
If the rest are incident on 238U there’s a chance of inducing
n+ U  U + 
238
239
U
1/2= 25 min
b-
239
239
Np
1/2= 2.3 days
b-
239
A well designed breeder reactor
can double the amount of fissile material
in 7 – 10 years.
Pu