PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
CH.I : INTRODUCTION TO REACTOR
PHYSICS
FROM THE FISSION PROCESS TO THE
REACTOR CHARACTERISTICS DEFINITION
•
•
•
•
NUCLEAR FUELS
n – heavy nucleus INTERACTION
NEUTRON CYCLE AND CRITICALITY
CONSTITUTIVE ELEMENTS OF REACTORS
CROSS SECTIONS PROFILES
• INTERACTION MECHANISMS
• ASSOCIATED CROSS SECTIONS
NUCLEAR FISSION
• DESCRIPTION
• DELAYED NEUTRONS
1
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
I.1 FROM THE FISSION PROCESS TO
THE REACTOR CHARACTERISTICS
DEFINITION
NUCLEAR FUELS
Energy production by nuclear fission
Stability of heavy nuclides
 Excess of neutrons (n) with respect to protons
 Last stable element in natural conditions: U (Z=92)
Natural distribution of U isotopes:
•
•
•
U28 (A=238) : ~ 99,3 %
U25 (A=235) : ~ 0,7%
U24 (A=234) : traces
2
NUMBER OF NEUTRONS (N)
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Stable nuclides – N-Z relation
3
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Binding energy B(A,Z) of the nuclei
Default of mass of a nucleus:  between  masses of
its constituents and its own mass
 Maximum about A~50
Possible release of energy either by fusion of light
nuclei or fission of heavy nuclei
Yet a spontaneous process?
4
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Fissile and fertile isotopes
Nuclear reactor  possibility to fission nuclei
– by neutron bombing,
– with production of additional neutrons,
– and chain reaction self-sustained
Few possible isotopes (called fissile):
U235 (natural), U233, Pu239
Fertile isotopes: neutron capture  fissile isotopes
U238 + n
Th232 + n


U239 + 
Th233 + 
23’
2.3 j
 Np239

-
22’
 Pa233
-
-
Pu239
27.4 j

-
U233
5
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Fission energy
1 nucleus U235
fission
200 MeV
1 atom C
combustion
3 eV
(same order of magnitude for other fissile isotopes)
1g U235  200 MeV x NA / 235  8.2 x 1010 J
 ~ 1 MWj !
(some losses however + U235 far from being totally consumed in
current PWRs)
6
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
•
Thermal power – MW thermal (MWth)
Energy produced by the reactor / unit time
•
Electrical power – MW electric (MWe)
output of the alternator
•
Efficiency of the thermodynamic cycle: ~ 33%
•
Power of PWR reactors:
900 (3 loops) or 1300 (4 l.) Mwe
 Consumption of fissile material / day = …
BUT: natural abundance of U isotopes unfavorable!
 Artificial enrichment of the nuclear fuel in U235 to reach a
sufficient power density (for light water reactors)
7
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
‘n – heavy nucleus’ INTERACTIONS
Possible phenomena
•
Scattering due to collisions
elastic
inelastic
•
Fission:
U25 + n  2 fragments +  n +  and  radiations
( : nb of n / fission)
•
Radiative capture:
U25 + n  U26 + 
•
…
8
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Cross section
Proba of those interactions?
Macroscopic cross section 
interaction proba of a n per unit length of its free
flight in a media
 : [cm-1]
function of




the media
the energy of the n (i.e. relative velocity v between n –
nucleus)
the spatial position (if variation in the isotopic composition
of the material)
the interaction type
9
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Cross sections considered in reactor physics
in
Inelastic scattering
e
Elastic scattering
s = e +in
Total scattering
f
c
 a =  c + f
Fission
Absorption (capture + fission)
 t =  a + s
All interactions
 capture
(Q: type of proba density function (pdf) for the free flight?)
For an isotope,   N, isotopic density ([cm-3])

 = N.
f(p,T)
f(nucleus, v)
 : microscopic cross section [cm2]
10
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Illustrative interpretation of : ‘visible’ cross section of
a nucleus for a n of velocity v
 More appropriate unit: 1 barn = 10-24 cm2
Dependence of  as a function of energy E
( proba of a particular interaction)
See next slides
Energy ranges to be accounted for:
 Energy of the n produced by fission: O(MeV)
 Energy of the n after their slowing-down due to
successive collisions, i.e. thermal (thermal equilibrium
with the media) : 0.025 eV
 8 decades !
11
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
See chap. VI
12
13
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
http://www.world-nuclear.org/education/phys.htm
14
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Behavior of f (U235)
 Fissions much more probable at low energy
 Numerous resonances at intermediate energies
 Thermalisation of the n
 Efficient slowing-down of the n (with a material with a
low A, e.g. H2)
 Absorptions avoided (low capture proba)
 Possible moderator : H20 (also cooling fluid)
 Thermal reactors
15
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Behavior of f (U238)
 Fissile at high energy (> 0.8 MeV)
 f (U238) << f (U235)
But possible production of Pu239, fissile with a high 
Breeding ratio:
BR =
Production rate of fissile material
Destruction rate of fissile material
BR > 1: conversion  breeding
 Interest of fast reactors
(reconsidered in Gen-IV)
But: n slowing-down to be avoided
 Coolant: molten metals (Na!), molten salts…
16
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
NEUTRON CYCLE AND CRITICALITY
Cycle characteristics
1. Expected number of n / fission: (E)
For U235 :
2.432  0.066 E
 (E)  
 2.349  0.15E
E 1
E 1
(E in MeV)
17
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
2.
Regeneration factor
=
Nb n produced in the fuel
Nb n absorbed in the fuel
 f



 f   c 1 
with  = c / f
Strong dependence on E !
 For U235 thermal n:  = 2.07
 For Unat thermal n:  = 1.34
 of  in the fast domain because  
Mixture of isotopes:
 


 (  
j
j
j
fj
fj
cj
)
18
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
3. Breeding ratio:
•
•
BR = ( - 1).F
if F = fraction of the fission n absorbed in fertile material
because  - 1 : excess of n available for breeding
Distribution of the reaction products
n + fissile isotope
 / (1 + ) isotope resulting from
capture (e.g. U236 for U235)
+ 2 / (1 + ) fission products
+n
(+ , radiations)
19
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Criticality
Reactor in regime mode

nb n produced = nb n consumed

chain reaction exactly self-sustained
Multiplication factor of a reactor: k
= expected nb of n emitted during a cycle/initial n
Rules the evolution of the neutron population:
 k < 1 : sub-critical reactor
 k = 1 : critical reactor
 k > 1 : super-critical reactor
20
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Expression of the multiplication factor
(case of a thermal reactor with uranium)
Infinite reactor: k
E
O(MeV)
 fission n
(fast) 0.8MeV
1 thermal n
absorbed in
the fuel
 n
Fast fissions
in U238
p n
fission
pf n
Slowing-down
0.025 eV
Absorption of
the thermal n
in the fuel
 : fast fission factor
p : resonance escape probability
f : thermal usage factor
k = pf
Typical values:
 = 1.65
 = 1.02
p = 0.87
f = 0.71
21
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Finite reactor: keff
leakages !!!
• Pf : non-leakage proba during slowing-down
• Pth : non-leakage proba at thermal energy
keff = k.PfPth = pfPfPth
P = Pf Pth < 1  criticality  k > 1
P, k : f(geometry and nature of materials, enrichment)
But leakages dependent on the surface/volume ratio
 Critical size (hence mass) of the reactor
 Balance between geometry and neutronics properties
Leakage reduction: reflector around the reactor
Reactivity: “distance to criticality”
Main protection: neutron-absorbing control rods
Emergency shutdown (“scram”): control rods dropped between
fuel rods
22
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
CONSTITUTIVE ELEMENTS OF REACTORS
Element
Objective
Example
Fissile isotopes
Provide the energy from
fissions
U235, U233, Pu239
Fertile isotopes
Convertible in fissile isotopes
U238, Th232
Fission energy
Thermal  fast
Thermal reactors 
breeders (breeders)
Moderator
Slow down fission n in thermal
n
H2O, D2O, graphite
Coolant
Cool down the core and
transport the energy produced
by fissions
H2O, D2O (thermal)
Na, molten metals (fast)
Thermodynamic
cycle
Pressurized (PWR) or
boiling (BWR) water
23
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Element
Objective
n-absorbing
materials
Control reactivity
Reflector
Limit leakages (located around
the core, same properties as
moderator)
Shielding
Reduce losses and doses
Example
Control rods
Thermal and biological
walls
…
•Many reactor designs were proposed
•Economical interest of some of them only (PWR, BWR…)
•Technological difficulties met for some designs
(see Na technology for fast breeders when first tried)
•New or more developed designs under study
24
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
I.2 CROSS SECTION PROFILES
INTERACTION MECHANISMS
 associated to the n  nucleus size (in the E range
considered)
 Interaction with the nucleus as a whole
Two mechanisms:
 Scattering of potential : « shocks » without interaction
with the internal structure of the nucleus
 Resonances  interaction with the internal structure:
 Constitution of a composed nucleus
 Deexcitation
Eexcitation = f(Ebinding(n),Ekinetic(n))
25
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Energy carried by an incident n on a nucleus
• System 1: n incident (with Ecin) on a (A,Z) nucleus
 Total energy of the system: Etot=m(A,Z)c2+mnc2+Ecin
• System 2: Compound nucleus (A+1,Z)
 Total energy of the system: Etot=m(A+1,Z)c2+Ecin+…?
Binding energy of nucleus (A,Z)
 Energy associated to the mass default of the nucleus:
B(A,Z)=(A-Z)mnc2+Zmpc2-m(A,Z)c2
Separation energy of a n out of nucleus (A+1,Z)
 Energy to be provided to the n to escape from the nucleus:
Sn=B(A+1,Z)-B(A,Z)=mnc2+m(A,Z)c2-m(A+1,Z)c2
Therefore: Etot=m(A+1,Z)c2+Ecin+Sn
 As if the incident n was bringing Ecin + Sn to the compound
nucleus
26
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
ASSOCIATED CROSS SECTIONS
 capture
deexcitation
n absorbed
composed nucleus

  : resonances at the E levels of the composed kernel
 Breit-Wigner at level Eo (for well-separated levels)
 

k
2
gJ
n 

( E  Eo )  4
2
k : wave nb  k2  E
2
1
 o
 1 x2
x = (E-Eo) / (/2)
(relative Ekin of the n – nucleus system)
gJ : statistical spin factor
, n,  (=  + n) : peak width
(capture, scattering and total, resp.)
27
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Behavior of 
 indep. of E and n  E
• E<<Eo :  ~ 1 / E
• E>>Eo :  ~ 1 / E5/2
 region in 1/v
  
 max = o.( / ) ~ 1 / Eo  resonances smaller and smaller
and less and less separated
28
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Scattering
Scattering of potential
Elastic scattering of the n without composed kernel
 pa = 4R2
for E  [1eV,1MeV]
(R = radius of the nucleus)
Elastic scattering of resonance
n absorbed
composed nucleus
n reemitted with
“the same” E
(i.e. E of the c.o.m. conserved !!)
 s: similar shape to a capture resonance + scattering of
potential + interference
s 

k
2
gJ
n 
( E  Eo ) 
2
2
4
n ( E  Eo )
4R
2

gJ
 4R
2
2
k
( E  Eo )  4
interference
29
30
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Inelastic scattering
n absorbed
composed nucleus
n reemitted at
E < Ei
+ deexcitation of the nucleus by  emission
Condition:
n with sufficient Ei to excite the 1st level of the nucleus
 in = 0 up to a threshold (~ 10 keV for heavy nuclei)
Rem: threshold much higher for light nuclei  not involved in
the inelastic scattering taking place in a reactor
 What about moderation by light nuclei (e.g. H2) ?
By elastic collisions in the c.o.m. (center-of-mass) system !!
 Relative E conserved
 Efficient slowing-down of the n
31
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Inelastic scattering
32
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Fission
n absorbed
composed nucleus
n reemitted
+ deexcitation of the nucleus by fragmentation in 2 (or 3) lighter
nuclei
Possible threshold (no threshold  fissile nucleus)
Profile of f ~  : region in 1/v, then resonances less and less
separated, but then limited variations
33
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
I.3 NUCLEAR FISSION
DESCRIPTION
Energetic feasibility
Separation energy Sn of a n
S n  B( A, Z )  B ( A  1, Z )

B ( A, Z )
 B ( A, Z ) B( A  1, Z ) 
 ( A  1).


A
A 1 
 A
<0
(B(A,Z) : binding E
of the nucleus (A,Z))
 Sn < average binding E per nucleon for a heavy nucleus
(see. Graphic § I.1)
But Sn: minimum E to provide to a nucleus (A-1,Z) to form a
composed nucleus (A,Z)
34
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Semi – empirical formula for the nucleus mass
Heavy nuclei:
1st approx. B(Z,A)  A
Nucleus radius:

R = ro.A1/3
m(nucleus)  A

V(nucleus)  A
 Density +/- cst
 Nuclei ~ drops of incompressible liquid
 Semi-empirical formula of Weizsäcker :
Superficial
tension
Coulombian
repulsion
2
2
(
A

2
Z
)
Z
B( A, Z )  av A  as A2 / 3  aa
 ac 1/ 3   ( A, Z )
A
A
N – Z asymmetry
 a3p/ 4
 A
 ( A, Z )   0
 a p
 A3 / 4
Z and N even
A odd
Z and N odd
Spin parity factor
Dominant for
heavy nuclei
35
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Addition of a n to a nucleus: ‘brings’ Sn + Ekin
a. Nucleus (Z even, N odd)
 + ap / A3/4
nucleus (Z even, N even)
b. Nucleus (Z even, N even) 
 - ap / A3/4
nucleus (Z even, N odd)
Difference of Sn = 2. ap / A3/4
~ O(MeV) !
In case b., about 1 MeV of addittional Ekin necessary to excite
the composed nucleus from (A,Z)
Rem: case a. : U235, U233, Pu239  fissile
case b. : U238, Th232
 fertile
 fission threshold: 0.8MeV
36
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Spontaneous vs. induced fission
For
(A,Z)  (A1,Z1) + (A2,Z2)
We have m nucleus (A,Z) > i m nuclei (Ai,Zi)
(for large A,
see curve B(A,Z)/A)
 spontaneous fission? possible but not observed
See potential energy of the fragments as a function of d
Nuclear
forces
Coulombian
repulsion
Ec = Coulomb potential in
d = R1 + R2 between Z1 and Z2
charges
Ef = diff. binding E between
(A,Z) and the fragments
37
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Tunnel effect? Nuclei ‘a bit’ too heavy…
 E brought by n (Sn + Ekin) to overcome Ec - Ef
(or E of a )
Ex: symmetrical fission
Ef > Ec for A > ~ 260
Unstable nuclei
For A  [230,240]
Sn ~ 5 to 6 MeV
Ec – Ef ~ 5.5 to 6 MeV
 Induced fission possible
Rem: what about1st
divergence of a reactor?
38
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Development of an induced fission ((n,f) capture)
Absorption time of a n:  10-17 s
Lifetime of an excited composed nucleus: ~ 10-14 s
( fission n indep. of the charact. of the absorbed n)
Agitation of the nucleons in the excited nucleus
 Formation of 2 fragments
 Coulombian repulsion: ~ 10-20 s
(with various (Ai,Zi))
Unstable fragments (N/Z ratio outside equilibrium)
 Deexcitation by emission of prompt n (~ 10-17 s)
Spectrum of the emitted n (prompt fission spectrum):
+/- Maxwell distribution
 (E) 
2
E
 
.e

E
(isotropic spectrum)

(units?)
39
40
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Mass distribution
of the fission
fragments
41
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
• Eexcitation (fragments) : not fully consumed by n emission
 Emission of  (called prompt, ~ 10-14 s)
• Ekin (fragments) :
Larger part of the E released by fission
Lost per ionization and excitation of the atoms in the
media crossed
 Fragments  fission products
unstable because of lack of Z
disintegrations 
Remark:
~ 30 possible fission modes + disintegrations
 mix of ~ 180 different radioactive nuclei!
42
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
DELAYED NEUTRONS
Origin
Fission products in radioactive chains in an excited state:
 Usually  emission
 Sometimes (i.e. if Eexcitation sufficient) emission of n
 delayed neutrons and 
Emission time?
Linked to the half-lifetime of the previous isotope (precursor)
in the chain (because deexcitation time much shorter)
43
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Numerous precursors (> 50) but grouped in 6 classes
characterized by
o Half-lifetime Ti of the precursor
o Fraction I of fission neutrons in group i
o Average Ei with spectrum i(E)
( = ii = 0.68 %)
Distribution of the fission energy
Production delayed
Residual heat to be
evacuated even after
reactor shutdown
44
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
Evolution of the neutron population N
 : expected lifetime of a n / cycle ~ 10-4 s


dN keff  1

N
dt

N (t )  N (0).e
k eff 1

t
If keff = 1.001, we have N(1 s) / N(0) ~ e+10 !!!
(1   )  i iTi  101 s
But influence of the delayed n:  
 If keff = 1.001, we have N(1 s) / N(0) ~ e+0.01 !!!
 Delayed n compulsory for reactor control
 If keff = 1 / (1 - ) : criticality reached without delayed n
 prompt-critical threshold to avoid!!
45
PHYS-H406 – Nuclear Reactor Physics – Academic year 2015-2016
CH.I : INTRODUCTION TO REACTOR
PHYSICS
FROM THE FISSION PROCESS TO THE
REACTOR CHARACTERISTICS DEFINITION 
•
•
•
•
NUCLEAR FUELS
n – heavy nucleus INTERACTION
NEUTRON CYCLE AND CRITICALITY
CONSTITUTIVE ELEMENTS OF REACTORS
CROSS SECTIONS PROFILES 
• INTERACTION MECHANISMS
• ASSOCIATED CROSS SECTIONS
NUCLEAR FISSION 
• DESCRIPTION
• DELAYED NEUTRONS
46