IAEA/ICTP Workshop on:
Technology and Applications of Accelerator Driven
Systems (ADS)
Y. Kadi and A. Herrera-Martínez
CERN, Switzerland
October 17-28 2005, ICTP, Trieste, Italy
Y. Kadi / October 17-28, 2005
1
LECTURES OUTLINE
 LECTURE 1: Physics of Spallation and Sub-critical Cores:
Fundamentals
(Monday 17/10/05, 16:00 – 17:30)
 LECTURE 2: Nuclear Data & Methods for ADS Design I
(Tuesday 18/10/03, 08:30 – 10:00)
 LECTURE 3: Nuclear Data & Methods for ADS Design II
(Tuesday 18/10/03, 10:30 – 12:00)
 LECTURE 4: ADS Design Exercises I & II
(Tuesday 18/10/03, 14:00 – 17:30)
 LECTURE 5: Examples of ADS Design I
(Thursday 20/10/03, 08:30 – 10:00)
 LECTURE 6: Examples of ADS Design II
(Thursday 20/10/03, 10:30 – 12:00)
 LECTURE 7: ADS Design Exercises III & IV
(Thursday 20/10/03, 14:00 – 17:30)
Y. Kadi / October 17-28, 2005
2
Physics of Spallation & Sub-critical Cores:
Fundamentals
Y. Kadi
CERN, Switzerland
17 October 2005, ICTP, Trieste, Italy
Y. Kadi / October 17-28, 2005
3
Introduction to ADS
 The basic process of Accelerator-Driven Systems (ADS) is Nuclear
Transmutation (spallation, fission, neutron capture):
 First demonstrated by Rutherford in 1919 who transmuted
17O using energetic a-particles (14N + 4He  17O + 1p )
7
2
8
1
14N
to
 I. Curie and F. Joliot in 1933 produced the first artificial radioactivity
using a-particles (27AL13 + 4He2  30P15 + 1n0)
 The invention of the cyclotron by Ernest
O. Lawrence in 1939 (W.N. Semenov in
USSR) opened new possibilities:
 use of high power accelerators to
produce large numbers of neutrons
Y. Kadi / October 17-28, 2005
4
Introduction to ADS
 One way to obtain intense neutron sources is to use a hybrid
sub-critical reactor-accelerator system called AcceleratorDriven System:
 The accelerator bombards a target with
high-energy protons which produces a
very intense neutron source through the
spallation process.
 These neutrons can consequently be
multiplied in the sub-critical core
which surrounds the spallation target.
Y. Kadi / October 17-28, 2005
5
Historical Background
p The idea of producing neutrons by spallation with an accelerator has been
around for a long time:
+ In 1950, Ernest O. Lawrence at Berkeley proposed to produce plutonium
from depleted uranium at Oak Ridge. The Material Testing Accelerator
(MTA) project was abandoned in 1954.
+ In 1952, W. B. Lewis in Canada proposed to use an accelerator to
produce 233U from thorium, in an attempt to close the fuel cycle for
CANDU type reactors.
 Concept of accelerator breeder : exploiting the spallation
process to breed fissile material directly  soon abandoned.
 Ip ≈ 300 mA
+ Renewed interest in the 1980's and beginning of the 1990's, in particular
in Japan (OMEGA project at Japan Atomic Energy Research Institute),
and in the USA (Hiroshi Takahashi et al. proposal of a fast neutron hybrid
system at Brookhaven for minor actinide transmutation and Charles
Bowman a thermal neutron molten salt system based on the thorium
cycle at Los Alamos).
Y. Kadi / October 17-28, 2005
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The Energy Amplifier
p In November 1993, Carlo Rubbia proposed, in an exploratory phase, a first
Thermal neutron Energy Amplifier system based on the thorium cycle, with a
view to energy production. As it became clear that in the western world the
priority is the destruction of nuclear waste (other sources of energy are
abundant and cheap), the system evolved towards that goal, into a Fast
Energy Amplifier.
Y. Kadi / October 17-28, 2005
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Conceptual study of an Energy
Amplifier
Subcritical system driven by a proton
accelerator:
 Fast neutrons (to fission all
transuranic elements)
 Fuel cycle based on thorium
(minimisation of nuclear waste)
 Lead as target to produce neutrons
through spallation, as neutron
moderator and as heat carrier
 Deterministic safety with passive
safety elements (protection against
core melt down and beam window
failure)
Y. Kadi / October 17-28, 2005
8
Review of Existing ADS Concepts
p
Classification of existing ADS concepts according to their physical features
and final objectives:
Ref. IAEA-TECDOC-985
 neutron energy
spectrum
fuel form
(solid/liquid)
fuel cycle
coolant-moderator
type
final objectives
Y. Kadi / October 17-28, 2005
9
The Spallation Process (1)
p Several nuclear reactions are capable of producing neutrons
 However the use of protons minimises the energetic
cost of the neutrons produced
Deposited
Neutrons
Energy
Emmitted
Per Neutron
(n/s)
(MeV)
Nuclear
Reactions
Incident Particle
&
Typical Energies
Beam
Currents
(part./s)
Neutron
Yields
(n/inc.part.)
Targ et
Power
(MW)
(e,) & (,n)
e- (60 MeV)
5  1015
0.04
0.045
1500
2  1014
H2(tn)He4
H3 (0.3 MeV)
6  1019
10-4 — 10-5
0.3
104
1015
Fission
- 1
57
200
2  1018
Spallation
(non-fissile target)
14
0.09
30
2  1016
30
0.4
55
4  1016
p (800 MeV)
Spallation
(fissionable target)
Y. Kadi / October 17-28, 2005
1015
10
Properties of the spallation
induced secondary shower
 one can distinguish between two qualitatively different
physical processes:
 A spallation-driven high-energy phase , commonly exploited in
calorimetry
• Complex processes
• Cross sections not so well known
• Parametrized in an approximate manner by phenomenological
models and MonteCarlo simulations
 A low-energy neutron transport phase, dominated by fission
• Diversified phenomenology down to thermal energies
• Main physical process governed by neutron diffusion
• Neutrons are multiplied by fissions and (n,xn) reactions
 The high-energy neutrons produced by spallation act as a source for
the following phase, in which they gradually loose energy by
collisions. The phenomenology of the second phase recalls that of
ordinary reactors with however some major differences.
The presence of the second phase is essential for obtaining
the high gains in energy.
Y. Kadi / October 17-28, 2005
11
The Spallation Process (2)
p There is no precise definition of spallation  this term covers
the interaction of high energy hadrons or light nuclei (from a
few tens of MeV to a few GeV) with nuclear targets.
 It corresponds to the reaction
mechanism by which this high
energy projectile pulls out of the
target some nucleons and/or light
particles, leaving a residual nucleus
(spallation product)
 Depending upon the conditions,
the number of emitted light
particles, and especially neutrons,
may be quite large
 This is of course the feature of
outermost importance for the socalled ADS
Y. Kadi / October 17-28, 2005
12
The Spallation Process (3)
p At these energies it is no longer correct to think of the
nuclear reaction as proceeding through the formation of
a compound nucleus.
 Fast Direct Process:
 Intra-Nuclear Cascade
(nucleon-nucleon collisions)
 Pre-Compound Stage:
 Pre-Equilibrium
 Multi-Fragmentation
 Fermi Breakup
 Compound Nuclei:
 Evaporation (mostly neutrons)
 High-Energy Fissions
 Inter-Nuclear Cascade
 Low-Energy Inelastic Reactions
 (n,xn)
 (n,nf)
 etc...
Y. Kadi / October 17-28, 2005
13
The Spallation Process (4)
p The relevant aspects of the spallation process are
characterised by:
+ Spallation Neutron Yield (i.e. multiplicity of emitted neutrons)
 determines the requirement in terms of the accelerator
power (current and energy of incident proton beam).
+ Spallation Neutron Spectrum (i.e. energy distribution of
emitted neutrons)
 determines the damage and activation of the structural
materials (design/lifetime of the beam window and spallation
target, radioprotection)
+ Spallation Product Distributions
 determines the radiotoxicity of the residues (waste
management).
+ Energy Deposition
 determines the thermal-hydraulic requirements (cooling
capabilities and nature of the spallation target).
Y. Kadi / October 17-28, 2005
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Spallation Neutron Yield
p
The number of emitted neutrons varies as a function of the target
nuclei and the energy of the incident particle  saturates around 2 GeV.
p
Deuteron and triton projectiles produce more neutrons than protons
in the energy range below 1-2 GeV  higher contamination of the
accelerator.
Y. Kadi / October 17-28, 2005
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Spallation Neutron Spectrum
p The spectrum of spallation neutrons evaporated from an
excited heavy nucleus bombarded by high energy
particles is similar to the fission neutron spectrum but
shifts a little to higher energy  <En> ≈ 3 – 4 MeV.
Y. Kadi / October 17-28, 2005
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Spallation Product Distribution
p The spallation product distribution
varies as a function of the target
material and incident proton
energy.
It
has
a
very
characteristic shape:

At
high
masses
it
is
characterized by the presence of
two peaks corresponding to
(i) the initial target nuclei and (ii)
those obtained after evaporation
 Three very narrow peaks
corresponding to the evaporation
of light nuclei such as (deuterons,
tritons, 3He and a)

An
intermediate
zone
corresponding to nuclei produced
by high-energy fissions
Y. Kadi / October 17-28, 2005
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Energy Deposition
p
Example of the heat deposition of a proton beam in a beam window and a Lead target
 which takes into account not only the electromagnetic interactions, but all
kind of nuclear reactions induced by both protons and the secondary generated
particles (included neutrons down to an energy of 20 MeV) and gammas.
 Increasing the energy of the incident particle
affects considerably the power distribution in the
Lead target. Indeed one can observe that, while
the heat distribution in the axial direction
extends considerably as the energy of the
incident particle increases, it does not in the
radial direction, which means that the proton
tracks tend to be quite straight.  Lorentz boost
 Heat deposition is largely contained within the
range of the protons. But while at 400 MeV the
energy deposit is exactly contained in the
calculated range (16 cm), this is not entirely true
at 1 GeV where the observed range is about 9%
smaller than the calculated (rcalc = 58 cm, robs ~
53 cm). At 2 GeV the difference is even more
relevant (rcalc = 137 cm, robs ~ 95 cm). This can
be explained by the rising fraction of nuclei
interactions with increasing energy, which
contribute to the heat deposition and shortens
the effective proton range.
Y. Kadi / October 17-28, 2005
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Models and Codes for HighEnergy Nuclear Reactions: FLUKA
Authors: A. Fasso1, A. Ferrari2,3, J. Ranft4, P.R.
Sala2,5
1
SLAC Stanford,
2
INFN Milan,
CERN,
Zurich
3
4
Siegen University,
5
ETH
Interaction and Transport Monte Carlo code
Web site: http://www.fluka.org
Y. Kadi / October 17-28, 2005
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FLUKA Description
 FLUKA is a general purpose tool for calculations of particle transport
and interactions with matter, covering an extended range of
applications spanning from proton and electron accelerator shielding
to target design, calorimetry, activation, dosimetry, detector design,
Accelerator Driven Systems, cosmic rays, neutrino physics,
radiotherapy etc.
 60 different particles + Heavy Ions
 Hadron-hadron and hadron-nucleus interaction 0-20 TeV
 Nucleus-nucleus interaction 0-1000 TeV/n [under development]
 Charged particle transport – ionization energy loss
 Neutron multi-group transport and interactions 0-20 MeV
 n interactions
 Double capability to run either fully analogue and/or biased
calculations
Y. Kadi / October 17-28, 2005
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FLUKA – Hadronic Models
Inelastic Nuclear Interactions
 Cross sections:
 Hadron-Nucleon
Parameterized fits for hadron–hadron
Tabulated data plus parameterized fits for hadron-nucleus
• 5 GeV - 20 TeV Dual Parton Model (DPM)
• 2.5 - 5 GeV
Resonance production and decay model
 Hadron-Nucleus
• < 4-5 GeV
PEANUT + Sophisticated Generalized
Intranuclear Cascade (GINC) pre-equilibrium
• High Energy
Glauber-Gribon multiple interactions Coarser
GINC
All models: Evaporation / Fission / Fermi break-up /
-deexcitation of the residual nucleus
Elastic Scattering
 Parameterized nucleon-nucleon cross sections.
 Tabulated nucleon-nucleus cross sections
Y. Kadi / October 17-28, 2005
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Hadron-Nucleon Cross Section
Total and elastic cross section for pp and p-n scattering, together with
experimental data
Isospin decomposition of p-nucleon
cross section in the T=3/2 and
T=1/2 components
Hadronic interactions are mostly surface effects  hadron nucleus cross
section scale with the target atomic mass A2/3
Y. Kadi / October 17-28, 2005
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Inelastic hN interactions
Intermediate Energies
 N1 + N2  N1’ + N2’ + p
 p + N  p’ + p” + N’
threshold around 290 MeV
important above 700 MeV
opens at 170 MeV
 Dominance of the D(1232) resonance and of the N* resonances  reactions
treated in the framework of the isobar model  all reactions proceed
through an intermediate state containing at least one resonance
 Resonance energies, widths, cross sections, branching ratios from data and
conservation laws, whenever possible
High Energies
 Interacting strings (quarks held together by the gluon-gluon interaction into
the form of a string)
 Interactions treated in the Reggeon-Pomeron framework
 Each colliding hadron splits into two color partons  combination into two
color neutron chains  two back-to-back jets
Y. Kadi / October 17-28, 2005
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PEANUT
PreEquilibrium Approach to NUclear Thermalization
 PEANUT handles hadron-nucleus interactions from threshold
(or 20 MeV neutrons) to 3-5 GeV
Sophisticated Generalized IntraNuclear Cascade
Smooth transition (all non-nucleons emitted/decayed + all
secondaries below 30-50 MeV)
Prequilibrium stage
Standard Assumption on exciton number or excitation energy
Common FLUKA Evaporation model
Y. Kadi / October 17-28, 2005
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hA at High Energies
Hadron-Nucleus interactions above 3-5 GeV/c
 primary interaction: Glauber-Gribon multiple interactions
 secondary particles: Generalized IntraNuclear Cascade;
Essential ingredient: Formation zone
 Last stage:
Common FLUKA evaporation module
Glauber cascade
 Elastic, Quasi-elastic and Absorption hA cross section derived
from Free hadron-Nucleon cross section + Nuclear ground
state only.
 Inelastic interaction = multiple interaction with v target
nucleons with binomial distribution
 A v
Pr ,v (b)    Pr (b) [1  Pr (b)] Av
v
Y. Kadi / October 17-28, 2005
25
Generalized IntraNuclear
Cascade
 Primary and secondary particles moving in the nuclear medium
 Target nucleons motion and nuclear well according to the Fermi gas
model
 Interaction probability
sfree + Fermi motion × r(r) + exceptions (ex. p)
 Glauber cascade at higher energies
 Classical trajectories (+) nuclear mean potential (resonant for p)
 Curvature from nuclear potential  refraction and reflection
 Interactions are incoherent and uncorrelated
 Interactions in projectile-target nucleon CMS  Lorentz boosts
 Multibody absorption for p, m-, K Quantum effects (Pauli, formation zone, correlations…)
 Exact conservation of energy, momenta and all addititive quantum
numbers, including nuclear recoil
Y. Kadi / October 17-28, 2005
26
Advantages and Limitations of
GINC
Advantages
 No other model available for
energies above the pion threshold
production (except QMD models)
 No other model for projectiles
other than nucleons
 Easily available for on-line
integration into transport codes
 Every target-projectile
combination without any extra
information
 Particle-to-particle correlations
preserved
 Valid on light and on heavy nuclei
 Capability of computing cross
sections, even when it is unknown
Y. Kadi / October 17-28, 2005
Limitations
 Low projectile energies
E<200MeV are badly described
 Quasi electric peaks above
100MeV are usually too sharp
 Coherent effect as well as
direct transitions to discrete
states are not included
 Nuclear medium effects, which
can alter interaction properties
are not taken into account
 Multibody processes (i.e.
interaction on nucleon clusters)
are not included
 Composite particle emissions
(d,t,3He,a) cannot be easily
accommodated into INC, but
for the evaporation stage.
 Backward angle emission
poorly described (Corrected for
FLUKA)
27
Residual Nuclei
 The production of residuals is the
result of the last step of the
nuclear reaction, thus it is
influenced by all the previous
stages
 Residual mass distributions are
very well reproduced
 Residuals near to the compound
mass are usually well reproduced
 However, the production of specific
isotopes may be influenced by
additional problems which have
little or no impact on the emitted
particle spectra (Sensitive to
details of evaporation, Nuclear
structure effects, Lack of spinparity dependent calculations in
most MC models)
Y. Kadi / October 17-28, 2005
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Electrons and Photons in FLUKA
Contents
 Electro Magnetic FLUKA (EMF) at a glance
 Physical Interactions
 Transport
 Biasing
Y. Kadi / October 17-28, 2005
29
Low Energy Neutron Transport in
FLUKA
Contents
 Multigroup technique
 FLUKA Implementation
 Cross section libraries and materials
 Energy weighting
 Other library features
 Possible Artifacts
 Secondary Particle production and transport
 Secondary Neutrons
 Gammas
 Fission Neutrons
 Charged Particles
 Residual Nuclei
Y. Kadi / October 17-28, 2005
30
Sub-Critical Systems
 In Accelerator-Driven Systems a Sub-Critical blanket
surrounding the spallation target is used to multiply the
spallation neutrons.
Y. Kadi / October 17-28, 2005
31
Sub-Critical vs Critical Systems
 ADS operates in a non self-sustained
chain reaction mode
 minimises criticality
and power excursions
 ADS is operated in a sub-critical mode
 stays sub-critical whether
accelerator is on or off
 extra level of safety against
criticality accidents
 The accelerator provides a control
mechanism for sub-critical systems
 more convenient than
control rods in critical reactor
 safety concerns, neutron
economy
 ADS provides a decoupling of the
neutron source (spallation source) from the
fissile fuel (fission neutrons)
 ADS accepts fuels that would not be
acceptable in critical reactors
 Minor Actinides
 High Pu content
 LLFF...
Y. Kadi / October 17-28, 2005
32
Reactivity Insertions
There is a spectacular
difference between a critical
reactor and an ADS
(reactivity in $ = r/b; r =
(k–1)/k) :
 Figure extracted from C. Rubbia et al.,
CERN/AT/95-53 9 (ET) showing the
effect of a rapid reactivity insertion in
the Energy Amplifier for two values of
subcriticality (0,98 and 0,96),
compared with a Fast Breeder Critical
Reactor.
 2.5 $ (Dk/k ~ 6.510–3) of reactivity
change corresponds to the sudden
extraction of all control rods from the
reactor.
Y. Kadi / October 17-28, 2005
33
Physics of Sub-Critical Systems
 The basic Physics describing the behaviour of neutrons in a sub-critical
system is identical to that of ordinary critical reactors. The general
properties of the flux are derived from the same equation which
expresses the principle of conservation of neutrons in a given system:
n
 Production  Absorption  Leakage
t
Corrected
for (n,xn)
ext. source
Production  n S f F  C (r ,t )
Absorption  S a F Fissions
(
)
Leakage   J   DF  D 2F
n = neutron density [n/cm3]; F = neutron flux [n/cm2/s]
n = neutrons emitted per fission; Sf = macroscopic fission cross section
= external neutron source (spallation neutrons for instance)
C(r ,t )
Sa = macroscopic absorption cross section(capture + fission)
= neutron current [n/cm2] according to Fick’s Law
J
(J  DF)
Y. Kadi / October 17-28, 2005
34
Physics of Sub-Critical Systems
1
1
1
 D is the diffusion coefficient :
D


3(St  Ssm) 3Str 3Ss
(high-A medium, little absorption)
m cos  0 for heavy nuclidesat low energies
n
 nS f F  C (r , t )  S a F  D 2F
t

 This equation holds only for mono-energetic neutrons, homogeneously
distributed in non absorbing medium, away from the source and
external boundaries
of the system. It is nevertheless almost valid in

the case of the Energy Amplifier since there is no strong absorption.
This equation enables us to understand the general characteristics of
the system.
 At Equilibrium (stationary solution) :
 The time dependence disappears, F et C are only functions of the space variables
n
S
C
 0   2F  (k 1) a F   0
t
D
D
 where k is defined as:
Y. Kadi / October 17-28, 2005
k 
nS f
Sa
(n,xn included in n)
35
Physics of Sub-Critical Systems
 The equation to be solved could be written :
 2F 
(k 1)F   C
L2c
D
(1)
D
2
where the diffusion length Lc is defined as : Lc  S
a

 The classical
way of solving this equation consist in finding a
general solution where the second term
of the equation is

set to zero:
 2 
(k 1)   0
L2c
(2)
it appears that there are two ways for the system to be subcritical (keff < 1), leading to two different sets of solutions:

 k > 1 : sub-criticality is obtained due to a lack of neutron
confinement, this is geometry related (EA : k ~ 1.2–1.3).
 k < 1 : the system is intrinsically sub-critical (FEAT : k ~ 0.93)
Y. Kadi / October 17-28, 2005
36

Material and Geometric Bucklings
 For a system of finite dimensions with k
>1:
2
2
 2   BM
  0; BM

(k 1)  nS f
L2c
 Sa
D
where B2M is referred to as the « material
buckling » (measure of the curvature).
B2M being positive means that the solution
is of oscillatory nature.
 Considering a finite system, with
vanishing flux at the (extrapolated)
boundaries, and a source also vanishing
at and outside the boundaries, we can
also write the solution in terms of the
eigenvectors of the characteristic "wave
equation" (B2i, where i is an integer ≥ 1,
also called « geometric buckling »).
 For a sub-critical system, the boundary
conditions, for a source with limited
extent, are less important than in critical
systems.
Y. Kadi / October 17-28, 2005
37
Material and Geometric Bucklings
 In a critical system, the condition for the flux to be everywhere
finite and non-negative, restricts the solution to the positive half of
the function obtained. B2i is therefore restricted to the lowest
eigenvalue, B21  B2G, that results from solving the wave equation:
 2   BG2   0
 B2M = B2G is the critical condition which expresses the equilibrium
between the geometrical and material component of the system
 In other words,the geometric buckling of a critical system of a specified
shape is equal to the material buckling for the given multiplying medium
 In a critical system of finite dimensions, the criticality condition is that
the effective multiplication factor shall be unity. In view of the critical
equation, the effective multiplication factor may be defined by:
2
k
DB M
k eff 
 1 where k  1 
2
Sa
DB M
/Sa  1
 DB2M/Sa represents the excess multiplication which is necessary to
compensate for the leakage of neutrons

Y. Kadi / October 17-28, 2005
38
Typical solutions
 The geometry of the system determines the type of oscillatory
solution. In a critical system, the fundamental mode alone is
important.
GEOMETRY
Fonction
(fondamental mode)
1 pr 
sin 
r  R 
Sphere of radius R
p 
 
R 
2
Cylinder of radius R,
height H,
centred at z=0
2,405r  pz 
J 0 
cos 
 R   H 
Parallelepiped of
sides A, B, C
centred at x=y=z=0

px  py  pz 
sin sin sin 
 A   B   C 

Y. Kadi / October 17-28, 2005
Geometrical
Buckling (B2i)

2,405   p 

   
  R  H 
2
2
p  p  p 
       
 A  B  C 
2
2
2
39
Study of a simplified sub-critical
system
 The general properties of a sub-critical system in the presence of
an external neutron source can be illustrated considering a
parallelepiped geometry:
 2   B 2   0 (2)
with the boundary condition that  (x)  0 in the planes x=a, y=b
and z=c, where a, b and c are the extrapolated distances. The
solutions of the wave equation (2) form a complete orthogonal set
of functions.Consequently, the space part of the neutron flux and

of the external neutron source
can be expanded in terms of an
infinite series of eingenfuncions  l,m,n :
 l,m,n (x) 


V
l,m,n
8
 px   py   pz 
sinl sinm sinn 
abc  a   b   c 

(x) l',m',n' (x)dV   (l  l') (m  m') (n  n')
with the following eigenvalues :
Y. Kadi / October 17-28, 2005
2
Bl,m,n
 l 2 m 2 n 2 
 p  2  2  2 
b
c 
a
2
40

Study of a simplified sub-critical
system
 The flux F and the external neutron source C can be expressed
as a linear combination of the eigenfunctions :
F( x ) 
F
l,m,n l,m,n
(x) where F l,m,n    l,m,n (x)F(x)dV
l,m,n l,m,n
1

D
l,m,n
C (x )  D
c
l,m,n
V
(x) where c l,m,n

V
l,m,n
(x)C (x)dV
 Introducing these expressions in the equation  2F B 2 F   C (1)
D
one can determine the coefficients Fl,m,n :
c
(k 1)
2
F l,m,n  2 l,m,n 2
where BM
 2
Bl,m,n  BM
Lc
the critical condition is given by the fact 
that : the
flux must be
non-zero whenever the coefficients cl,m,n tend towards zero. This
is only possible if k > 1, which is the case here. Therefore, the
 value of B2

smallest
in principle is equal to the
l,m,n , which
smallest value of k that makes the system critical, is obtained
for l = m = n = 1 (fundamental mode with sine distribution).
Y. Kadi / October 17-28, 2005
41
Leakage Probability
 The rate of absorption in a homogeneous volume V is given by:
Riabs 
  (x)S dV  S   (x)dV
V
i
a
a
V
i
 To calculate the rate of leakage for a given mode i = (l,m,n) :
one can multiply equation (2) by Sa, integrate over the volume
and use the definition of the leakage probability such that:


 2 i (x)S a dV  Bi2  i (x)S a dV  Bi2 Riabs


 2 i (x)dV  S a  i ( x)dV  
V
Using the divergence theorem one can rewrite the first term :
Sa

V

V
Sa
D

V
 J i (x )dV
S
1
  a J i (x)  dS   2 Rileak where J i ( x)  D i (x)
D S
Lc
The relation between leakage
and absorption rate is given by:


Rileak  RiabsL2c Bi2
B2i:
(3)
This illustrates the role of
for a given volume the leakage
 probability increases with the mode.

Y. Kadi / October 17-28, 2005
42
Leakage Probability
 Leakage and non-leakage probabilities:
Pi
leak
Rileak
L2c Bi2
 leak

;
Ri  Riabs 1  L2c Bi2
Pinonleak  1  Pileak 
where ki is the criticality factor for mode i : ki 


2
given that Bi2  Bl,m,n
1
ki

1  L2c Bi2 k
k
1  L2c Bi2
2
2
2 

l
m
n
 p 2  2  2  2  is a function that
b
c
a
 
is rapidly increasing with mode, escapes will be more important
the higher the mode. Therefore, one can deduce that if the
fundamental mode is sub-critical, then all the other modes will be
 more sub-critical.
even
 A new expression of the flux can be derived as a function of ki :
F l,m,n 
c l,m,n
2
2
Bl,m,n
 BM
c l,m,n
L2c c l,m,n
1



2
2
k 1 1  Bl,m,n
1  kl,m,n
2
L
c
Bl,m,n 
L2c
It is of interest to note the Amplification factor 1/(1-ki) specific to
every single mode.
Y. Kadi / 
October 17-28, 2005
43
Flux in a Sub-Critical System with
k > 1
 The general solution of the flux for a finite system with k > 1 is
given by:
F( x ) 
F
l,m,n l,m,n
l,m,n

(x ) 
L2c
1 B
c l,m,n
l,m,n
2
2
L
l,m,n c

 l,m,n (x)
1  k l,m,n
 Clearly k1 is higher than kn for n > 1. This implies that when the
reactivity of the system is progressively increased, and k1
approaches 1, the first term of the expansion of the flux diverges
[1/(1-k)] whereas the other terms remain finite and can be
neglected. (In the presence of the fundamental mode the flux will
have a finite amplitude since the capture rate will be precisely
adjusted in order to maintain the chain reaction).
 Without the external source, the higher harmonics of the system
n >1 will not be excited, and the multiplication factor of the subcritical system is therefore keff = k1. This is what is happening when
the proton beam is switched off in the energy amplifier.
Y. Kadi / October 17-28, 2005
44
Neutron multiplication in a subcritical system
 In an accelerator driven, sub-critical fission device  the "primary"
(or "source") neutrons produced via spallation initiate a cascade
process. The « source » neutrons are multiplied by fissions and
(n,xn) reactions through the multiplication factor M :
k n 1 1
M  1  k  k  k  ...  k 
k 1
2

3
n
n


1
1 k
for k  1
 If we assume that all generations in the cascade are equivalent, we
can define an average criticality factor k (ratio between the neutron
population in two subsequent generations), such that :
k
M 1
1
1
1
M
M
 From the previous discussion, it is clear that in the presence of a
source, k ≠ keff. We will indicate hereon with ksrc the value of k
calculated from
 the net multiplication factor M in the presence of an
external source.
k src
Y. Kadi / October 17-28, 2005
M 1

M
45
Calculation of the multiplication
factor
 By definition the neutron multiplication factor is given by :
F
abs
l,m,n Rl,m,n
leak
 F l,m,n Rl,m,n

where Q  C (x)dV
Q
V
the first term of the numerator corresponds to the rate of absorption
whereas the second term is related to the leakage of neutrons, for the
harmonic mode l,m,n. In other words, the total number of neutrons

produced
is equal to the sum of the neutrons absorbed (capture +
fission) and those which have leaked out of the system.
M
l,m,n
 By using equation (3) together with the definition of Fl,m,n we obtain:
L2c S a c l,m,n
abs
2 2
F l,m,n Rl,m,n 1  Lc Bl,m,n
 l,m,n ( x)dV
1

k
V
l,m,n
M  l,m,n
 l,m,n
Q
Q
hence :


M 

 
Dc l,m,n
M l,m,n

V
l,m,n

(x)dV
Q
l,m,n
and M l,m,n 
1
1  kl,m,n
 The Net Multiplication Factor is obtained by summing up the individual
factors of a given mode weighed by the source term corresponding to
that given mode.

QD
c
l',m',n'
Y. Kadi / October 17-28, 2005
l' m' n'

V
l',m',n'
(x)dV
46
Time Dependence
 The diffusion equation of neutrons in a non-equilibrium system
can be written as:
n( x, t ) 1 F(x,t )

 D 2F(x, t )  (k  1)S a F( x,t )  C ( x,t ) (4)
t
v
t
where v is the mean velocity of the neutrons, so that F = nv.
 
Consider the case of a neutron burst represented by C0(t), which
would correspond to a pulse of the accelerator. An attempt will be
made to solve this equation by separating the variables, I.e. by
setting:
F(x,t ) 
F
l,m,n l,m,n
(x) f l,m,n (t)
l,m,n
 Substituting in (4), following the properties of l,m,n lead to :
1 df (t )

2
 DB l,m,n
 (1  k )S a f (t) F l,m,n l,m,n (x)  0
dt

l,m,n
 v


this 
implies that the coefficient of every mode must be zero.

Y. Kadi / October 17-28, 2005
47
Time Dependence
 Every mode has therefore its own time dependence, solution of
the following equation :
df (t)
2
 v DB l,m,n
 (1  k )S a dt
f (t)
where :
2
2
v DBl,m,n
(1k )S a t
vS a 1L2c B l,m,n
(1kl,m,n )t

f (t)  e




the flux can then be expressed as :
F(x,t ) 


F l,m,n l,m,n (x)e
e
(
(
)
)
2
vS a 1L2c B l,m,n
(1kl,m,n )t
l,m,n
 Every single mode decreases therefore with its own time constant
which becomes shorter the higher the order of the mode. At
criticality (k1,1,1 =1), the term of the exponential is equal to zero
and the harmonic is infinitely long. Fermi was the first in using
the time evolution of a neutron pulse in order to be able to
control the approach to criticality of his reactor at Chicago in
1942. In an Energy Amplifier driven by a CW cyclotron, one
could use such a method by simply interrupting the proton beam
fort short periods (Jerk).
Y. Kadi / October 17-28, 2005
48
Neutronic characteristics of a
system intrinsically sub-critical
 In a multiplying medium made of natural uranium and water,
such as the one used in FEAT, k < 1. The diffusion equation in
which the system is in a steady state is expressed by :
 2F 
(k 1)F   C
L2c
D
(1)
 Consider a point source located at the centre of an infinite
homogeneous diffusion medium, with the result that in this

system the neutron
distribution will have spherical symmetry.
Expressing the Laplacian operator in spherical coordinates gives:
d 2F(r) 2 dF(r) 1  k


F(r)  0
r dr
dr 2
L2c
 Let u/r = F(r), the equation reduces to :

d 2u
1  k
2


u

0.
where


dr 2
L2c
remember that 1– k > 0.

Y. Kadi / October 17-28, 2005
49
Neutronic characteristics of a
system intrinsically sub-critical
 Since 2 is a positive quantity, the general solution is thus:
r
r
u  Ae  Ce
e r
er
FA
C
r
r
and hence
it is apparent that C must be zero, for otherwise the flux would become
infinite as
r  ∞, so that only A remains to be determined. The neutron
current density at a point r is given by:

J (r)  D
dF(r )
dr
(
)
(Fick' s law) and Q  limite 4pr 2 J (r )
r0
upon inserting the value for A, it follows that

F(r ) 
Q
e r
4 pDr
 The value of  depends on the physical properties of the sub-critical
assembly considered. The important characteristic is the exponential
decrease of the flux as a function of distance. In the case of a spallation
source which is
not pointlike, this behaviour will only be valid at a certain
distance from the centre of the source (a few collision lengths away [lc =
3 cm in Pb for instance]).
Y. Kadi / October 17-28, 2005
50
Particular case of FEAT
 A more detailed theory shows that in order to account more
efficiently for the escapes of fast and thermal neutrons, one should
replace the diffusion length by the migration length, introducing
thus the quantity called Fermi age (t) :
M L t
where t (E ) 
hence : a
1 k
M c2
2
c
2
c


E0
E
D
dE
Ss E
and  the average letharg y
 If we take k = 0.93 and M2c = 30 cm2 as in FEAT, one finds that
 = 0.0483 cm–1 , i.e. quite close to the measured value of 0.0458
cm–1.
(1/  ~ 21 cm)
 For water k = 0 et  = 0.020 cm–1 !
(1/  ~ 5.5 cm)
Y. Kadi / October 17-28, 2005
51
Spatial distribution of the
neutron flux
Y. Kadi / October 17-28, 2005
52
The FEAT (First Energy Amplifier
Test) Experiment 1993-94
EXPERIMENTAL DETERMINATION OF THE ENERGY GENERATED BY NUCLEAR
CASCADES FROM A PARTICLE BEAM
CEN, Bordeaux-Gradignan, France
CIEMAT, Madrid, Spain
CSNSM, Orsay, France
CEDEX, Madrid, Spain
CERN, Genève, Switzerland
Dipartimento di Fisica e INFN, Università di Padova, Padova, Italy
INFN, Sezione di Genova, Genova, Italy
IPN, Orsay, France
ISN, Grenoble, France
Sincrotrone Trieste, Trieste, Italy
Universidad Autónoma de Madrid, Madrid, Spain
Universidad Politecnica de Madrid, Madrid, Spain
University of Athena, Athens, Greece
Université de Bâle, Bâle, Switzerland
University of Thessalonic, Thessalonique, Greece
Y. Kadi / October 17-28, 2005
53
FEAT
 Top and side view of the FEAT assembly, on the T7
beam line of the CERN-PS complex.
Y. Kadi / October 17-28, 2005
54
The FEAT Assembly
 3.6 tonne of natural uranium immersed in water
Y. Kadi / October 17-28, 2005
55
Properties of the spallation
induced secondary shower
 one can distinguish between two qualitatively different
physical processes:
 A spallation-driven high-energy phase , commonly exploited in
calorimetry
• Complex processes
• Cross sections not so well known
• Parametrized in an approximate manner by phenomenological
models and MonteCarlo simulations
 A low-energy neutron transport phase, dominated by fission
• Diversified phenomenology down to thermal energies
• Main physical process governed by neutron diffusion
• Neutrons are multiplied by fissions and (n,xn) reactions
 The high-energy neutrons produced by spallation act as a source for
the following phase, in which they gradually loose energy by
collisions. The phenomenology of the second phase recalls that of
ordinary reactors with however some major differences.
The presence of the second phase is essential for obtaining
the high gains in energy.
Y. Kadi / October 17-28, 2005
56
Simluation of FEAT
 Example of a secondary shower produced by a single proton
Y. Kadi / October 17-28, 2005
57
Detectors used in FEAT
 The aim was to measure (1) the fission rate distribution (3
different techniques) as well as (2) the heat deposition inside
the entire device (thermometers).
Detector
Active Element
Gas Ionisation
Chamber
4 ata Argon
Cir cular window , =15 mm
vertical plane
1 mg/cm2
deposit
Polycrystalline
Si Diode
thickness 300mm
Polycrystalline
Si Diode
thickness 300mm
Thermistances
in metallic probes
Lexan foils track
detectors
Rectangu lar. window
9.6 x 11.6 mm 2
vertical plane
1 mg/cm2
deposit
90° sector
r = 14 mm
horizon tal plane
U Cyli nde rs : =8 mm
Pb Cylind er : =10 mm
Equi la teral t riang le s r =37mm
Cir cles =32 mm
Rectang le 10x25 mm 2 (Vert. Plane )
Y. Kadi / October 17-28, 2005
Uranium
converter
Clustering
Location
1 array of 6, spaced
10 cm ve rticall y .
2 arrays of 8, spaced
12.3 cm ver tically
10 arrays of 16
coun ters, spaced
6.4 cm ver tically
in water,
between U bars
1 mg/cm2
deposit
4 clusters of 6
coun ters spaced
21.3 cm ver tically
in fuel bar,
between U
cartridg es
~ 55 g U
3 thermometers wit h
two U probes
in water,
between U bars
5 ver tical s ets of 2
detectors
in water,
between U bars
~
1 mg/cm2
in water,
between U bars
58
Determination of the
multiplication factor k
 Several methods were used to determine k (source jerk, time
dependence, delayed neutron fraction, MC) : k = 0.895 ± 0.01
Y. Kadi / October 17-28, 2005
59
Flux measurement
 The neutron flux is parametrized according to an exponential function :
F(x,y,z) = const.  exp(–d/l) where the beam is along the x-axis, and
d = [(ax + )2 +y2 +z2]1/2 where  is a constant accounting for the average
axial displacement of the spallation shower with the beam energy and a is
the first moment of the longitudinal distribution of the shower.
Y. Kadi / October 17-28, 2005
60
Determination of the flux
parameters
 l is independent of the beam energy whereas a decreases slightly
when the beam energy increases.
Y. Kadi / October 17-28, 2005
61
Flux measurement
 The different types of detectors give coherent results. The energy
gain is calculated by integrating the fission density over the volume
of the device.
Y. Kadi / October 17-28, 2005
62
Measurement of the neutron flux
in FEAT
 Comparison between different Monte Carlo simulations
Y. Kadi / October 17-28, 2005
63
Measurement of the Energy Gain
 Behaviour of the measured gain matches MC simulation  FEAT
good benchmark for validating calculation methods and nuclear
data.
Y. Kadi / October 17-28, 2005
64
Simulation
 Innovative simulation tool based on MonteCarlo technique :
 Fluka for spallation and high-energy transport (En ≥ 20 MeV)
 EA-MC for low-energy neutron transport and time-evolution of the
material composition
 Most complete and detailed nuclear data bases: 800 nuclides with
reaction cross sections out of which ≈ 400 have also the elastic cross
section
 Complex geometry handling
 Use of special techniques (parallelisation, kinematics, cross-section
handling, etc…) to allow for high statistics (20 ms /event/CPU)
 such a complex tool need to be thoroughly validated (FEAT,
TARC, IAEA Benchmarks)
Y. Kadi / October 17-28, 2005
65
Conclusions from FEAT
 Consistant measurement of the energy gain.
 Validation of innovative MC simulation tool.
 Energy gain increases with particle beam energy 
constant above 900 MeV  modest requirement for
Energy Amplifier.
 The first Energy Amplifier (with a power rating ≈
watt) was operated at CERN in 1994.
Y. Kadi / October 17-28, 2005
66
Review of Sub-Critical Core
Experiments
p
Highly specified experiments have been carried out to verify the
fundamental physics principle of Accelerator-Driven Sub-Critical
Systems:
 The First Energy Amplifier Test (FEAT): S. Andriamonje et al.
Physics Letters B 348 (1995) 697–709 and J. Calero et al. Nuclear
Instruments and Methods A 376 (1996) 89–103;
 The MUSE Experiment (MUltiplication de Source Externe): M.
Salvatores et al., 2nd ADTT Conf., Kalmar, Sweden, June 1996;
 The YELINA Experiment (ISTC-B-70): S. Chigrinov et al.,
Institute of Radiation Physics & Chemistry Problems, National
Academy of Sciences, Minsk, Belarus.
Y. Kadi / October 17-28, 2005
67
The MUSE Experiment
MASURCA facility
(courtesy of CEA)
The Pulsed Neutron Source
« GENEPI »
Y. Kadi / October 17-28, 2005
68
The MUSE Experiment
Y. Kadi / October 17-28, 2005
69
Main MUSE Results
MUSE 1
MUSE 2
MUSE 3
MUSE 4
12/95
09-12/96
01-04/98
11/99-08/04
Cf252
Cf252
(D,T) thermalised
GENEPI
Stochastic
Stochastic
Pulsed
Pulsed
Apparent worth
of the source
Φ*
Buffer : Na or
SS
Φ*
Spectrum source
Dynamic
measurements
Spectrum index
(test)
Dynamic
measurements
M.A. Fission rates
Neutron
spectrometry
Lead zone
Y. Kadi / October 17-28, 2005
70
The YALINA Experiment
General view of the YALINA
fuel subassembly.
Y. Kadi / October 17-28, 2005
71
The YELINA Experiment (2)
Y. Kadi / October 17-28, 2005
72
The YELINA Experiment (4)
Experiment &
calculations:
keff vs. fuel load
Y. Kadi / October 17-28, 2005
73
Main YALINA Results
Neutron pulses measured by 3He-counters in different experimental channels.
–Close to criticality: straight line, α
constant
–Deep subcriticality: α time dependent
Y. Kadi / October 17-28, 2005
Layout of the Yalina core
74
Physics Validation Sequence
CONFIG
FEAT
MUSE
YALINA
YALINA
TRADE
RACE
SAD
EA
SOURCE
SPALL
DD/DT
DD/DT
DD/DT
SPALL
-NUCL
SPALL
SPALL
Y. Kadi / October 17-28, 2005
KINETICS
THERMAL
FAST
THERMAL
FAST
THERMAL
THERMAL
FAST
FAST
POWER EFFECTS
NO
NO
NO
NO
YES
NO
NO
YES
75