y. kadi / october 17-28, 20051 y. kadi and a. herrera-martínez cern, switzerland october 17-28...
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Y. Kadi / October 17-28, 2005 1
Y. Kadi and A. Herrera-MartínezCERN, Switzerland
October 17-28 2005, ICTP, Trieste, Italy
IAEA/ICTP Workshop on:Technology and Applications of Accelerator Driven
Systems (ADS)
Y. Kadi / October 17-28, 2005 2
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 3
Y. KadiCERN, Switzerland
17 October 2005, ICTP, Trieste, Italy
Physics of Spallation & Sub-critical Cores: Fundamentals
Y. Kadi / October 17-28, 2005 4
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 14N to 17O using energetic -particles (14N7 + 4He2 17O8 + 1p1)
I. Curie and F. Joliot in 1933 produced the first artificial radioactivity using -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 5
Introduction to ADS
One way to obtain intense neutron sources is to use a hybrid sub-critical reactor-accelerator system called Accelerator-Driven 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 6
Historical Background
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 7
The Energy Amplifier
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 8
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 9
Review of Existing ADS Concepts
Classification of existing ADS concepts according to their physical features and final objectives:
neutron energy spectrumfuel form (solid/liquid)fuel cyclecoolant-moderator typefinal objectives
neutron energy spectrumfuel form (solid/liquid)fuel cyclecoolant-moderator typefinal objectives
Ref. IAEA-TECDOC-985
Y. Kadi / October 17-28, 2005 10
The Spallation Process (1)
Several nuclear reactions are capable of producing neutrons
However the use of protons minimises the energetic cost of the neutrons produced
NuclearReactions
Incident Particle&
Typical Energies
BeamCurrents(part./s)
NeutronYields
(n/inc.part.)
TargetPower(MW)
DepositedEnergy
Per Neutron(MeV)
NeutronsEmmitted
(n/s)
(e,γ)&(γ,n) e-(60Me )V 5×1015 0.04 0.045 1500 2×1014
H2( )tn He4 H3(0.3Me )V 6×1019 10-4—10-5 0.3 104 1015
Fission ≈1 57 200 2×1018
Spallation(non-fissil etarge)t
Spallation(fissionabl etarge)t
p(800Me )V 101514
30
0.09
0.4
30
55
2×1016
4×1016
Y. Kadi / October 17-28, 2005 11
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 12
The Spallation Process (2)
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 so-called ADS
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 so-called ADS
Y. Kadi / October 17-28, 2005 13
The Spallation Process (3)
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...
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 14
The Spallation Process (4)
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 15
Spallation Neutron Yield
The number of emitted neutrons varies as a function of the target nuclei and the energy of the incident particle saturates around 2 GeV.
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 16
Spallation Neutron Spectrum
1E-6
1E-5
1E-4
1E-3
1E-2
1E-1
1E+0
1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8
Energy (eV)
Fission Source
Spallation Source
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 17
Spallation Product Distribution
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 )
An intermediate zone corresponding to nuclei produced by high-energy fissions
Y. Kadi / October 17-28, 2005 18
Energy Deposition
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.
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 19
Models and Codes for High-Energy Nuclear Reactions: FLUKA
Authors: A. Fasso1, A. Ferrari2,3, J. Ranft4, P.R. Sala2,5
1 SLAC Stanford, 2 INFN Milan, 3 CERN, 4 Siegen University, 5 ETH Zurich
Interaction and Transport Monte Carlo code
Web site: http://www.fluka.org
Y. Kadi / October 17-28, 2005 20
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 interactions Double capability to run either fully analogue and/or biased
calculations
Y. Kadi / October 17-28, 2005 21
FLUKA – Hadronic Models
Inelastic Nuclear InteractionsInelastic Nuclear Interactions Cross sections:
Hadron-NucleonParameterized fits for hadron–hadronTabulated 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 /γ-deexcitation of the residual nucleus
Elastic ScatteringElastic Scattering Parameterized nucleon-nucleon cross sections. Tabulated nucleon-nucleus cross sections
Y. Kadi / October 17-28, 2005 22
Hadron-Nucleon Cross Section
Total and elastic cross section for p-p 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 23
Inelastic hN interactions
Intermediate EnergiesIntermediate Energies N1 + N2 N1’ + N2’ + threshold around 290 MeV
important above 700 MeV + N ’ + ” + N’ opens at 170 MeV
Dominance of the 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 EnergiesHigh 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 24
PEANUT
PPreEEquilibrium AApproach to NUNUclear TThermalization
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 25
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 cascadeGlauber 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
vAr
vrvr bPbP
v
AbP −−⎟⎟
⎠
⎞⎜⎜⎝
⎛≡ )](1[)()(,
Y. Kadi / October 17-28, 2005 26
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
free + Fermi motion × (r) + exceptions (ex. ) Glauber cascade at higher energies Classical trajectories (+) nuclear mean potential (resonant for ) Curvature from nuclear potential refraction and reflection Interactions are incoherent and uncorrelated Interactions in projectile-target nucleon CMS Lorentz boosts Multibody absorption for , -, 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 27
Advantages and Limitations of GINC
AdvantagesAdvantages 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
LimitationsLimitations 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,) cannot be easily accommodated into INC, but for the evaporation stage.
Backward angle emission poorly described (Corrected for FLUKA)
Y. Kadi / October 17-28, 2005 28
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 spin-parity dependent calculations in most MC models)
Y. Kadi / October 17-28, 2005 29
Electrons and Photons in FLUKA
ContentsContents EElectro MMagnetic FFLUKA (EMF) at a glance Physical Interactions Transport Biasing
Y. Kadi / October 17-28, 2005 30
Low Energy Neutron Transport in FLUKA
ContentsContents 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 31
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 32
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...
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 33
Reactivity Insertions
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 $ (k/k ~ 6.510–3) of reactivity change corresponds to the sudden extraction of all control rods from the reactor.
There is a spectacular difference between a critical reactor and an ADS (reactivity in $ = /; = (k–1)/k) :
Y. Kadi / October 17-28, 2005 34
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 = neutron density [n/cm3]; = neutron flux [n/cm2/s] = neutrons emitted per fission; f = macroscopic fission cross section
= external neutron source (spallation neutrons for instance)a = macroscopic absorption cross section(capture + fission)
= neutron current [n/cm2] according to Fick’s Law
€
∂n
∂t= Production − Absorption − Leakage
€
Production = ν Σ f Φ + Cr r , t( )
€
Absorption = Σ aΦ
€
Leakage =r
∇ ⋅r J =
r ∇ ⋅ −D
r ∇Φ( ) = −D∇2Φ
€
Cr r , t( )
€
rJ
€
(r J = −D
r ∇Φ)
ext. source
Fissions
Correctedfor (n,xn)
Y. Kadi / October 17-28, 2005 35
Physics of Sub-Critical Systems
D is the diffusion coefficient : (high-A medium, little absorption)
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, et C are only functions of the space variables
where k is defined as: (n,xn included in )
€
∂n
∂t= νΣ f Φ + C
r r , t( ) − Σ aΦ + D∇2Φ
€
∂n
∂t= 0 ⇒ ∇2Φ + k∞ −1( )
Σ a
DΦ +
C
D= 0
€
k∞ ≡νΣ f
Σ a
€
D=1
3 Σt −Σsμ ( )=
13Σtr
≈1
3Σs
€
≡ cosθ = 0 for heavy nuclides at low energies
Y. Kadi / October 17-28, 2005 36
Physics of Sub-Critical Systems
The equation to be solved could be written :
where the diffusion length Lc is defined as :
The classical way of solving this equation consist in finding a general solution where the second term of the equation is set to zero:
it appears that there are two ways for the system to be sub-critical (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)
€
∇2Φ +k∞ −1( )
Lc2 Φ = −
C
D(1)
€
Lc2 ≡
D
Σ a
€
∇2Ψ +k∞ −1( )
Lc2 Ψ = 0 (2)
Y. Kadi / October 17-28, 2005 37
Material and Geometric Bucklings
For a system of finite dimensions with k > 1 :
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" (B2
i, 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.
€
∇2Ψ + BM2 Ψ = 0; BM
2 ≡k∞ −1( )
Lc2 =
νΣ f − Σ a
D
Y. Kadi / October 17-28, 2005 38
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. B2
i is therefore restricted to the lowest eigenvalue, B2
1 B2G, that results from solving the wave equation:
B2M = B2
G 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:
DB2M/a represents the excess multiplication which is necessary to
compensate for the leakage of neutrons€
k eff =k∞
DBM2 /Σ a + 1
= 1 where k∞ = 1 +DBM
2
Σ a
€
∇2Ψ + BG2 Ψ = 0
Y. Kadi / October 17-28, 2005 39
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)
Geometrical Buckling (B2
i)
Sphere of radius R
Cylinder of radius R, height H,centred at z=0
Parallelepiped of sides A, B, C centred at x=y=z=0
€
1
rsin
πr
R ⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
J02,405r
R ⎛ ⎝ ⎜
⎞ ⎠ ⎟ cos
πz
H ⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
sinπx
A ⎛ ⎝ ⎜
⎞ ⎠ ⎟sin
πy
B ⎛ ⎝ ⎜
⎞ ⎠ ⎟sin
πz
C ⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
R ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
€
2,405
R ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
+π
H ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
€
A ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
+π
B ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
+π
C ⎛ ⎝ ⎜
⎞ ⎠ ⎟2
Y. Kadi / October 17-28, 2005 40
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:
with the boundary condition that 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 :
with the following eigenvalues :
€
∇2Ψ + B2Ψ = 0 (2)
€
ψ l,m,nr x ( ) =
8
abcsin l
πx
a ⎛ ⎝ ⎜
⎞ ⎠ ⎟sin m
πy
b ⎛ ⎝ ⎜
⎞ ⎠ ⎟sin n
πz
c ⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
Bl,m,n2 = π 2 l 2
a2 +m2
b2 +n2
c 2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
ψ rx ( ) = 0
€
ψ l,m,nV∫
r x ( )ψ l ',m',n'
r x ( )dV = δ l − l'( )δ m − m'( )δ n − n'( )
€
ψ l,m,n
Y. Kadi / October 17-28, 2005 41
Study of a simplified sub-critical system
The flux and the external neutron source C can be expressed as a linear combination of the eigenfunctions :
Introducing these expressions in the equation one can determine the coefficients l,m,n :
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 smallest value of B2
l,m,n , which in principle is equal to the smallest value of k that makes the system critical, is obtained for l = m = n = 1 (fundamental mode with sine distribution).
€
r
x ( ) = Φ l,m,nψ l,m,nr x ( )
l,m,n∑ where Φ l,m,n = ψ l,m,n
r x ( )Φ
r x ( )
V∫ dV
€
Cr x ( ) = D cl,m,nψ l,m,n
r x ( )
l,m,n∑ where c l,m,n =
1
Dψ l,m,n
r x ( )C
r x ( )
V∫ dV
€
∇2Φ +B2Φ = −C
D(1)
€
l,m,n =cl,m,n
Bl,m,n2 − BM
2
€
where BM2 ≡
k∞ −1( )
Lc2
Y. Kadi / October 17-28, 2005 42
Leakage Probability
The rate of absorption in a homogeneous volume V is given by:
To calculate the rate of leakage for a given mode i = (l,m,n) : one can multiply equation (2) by a, integrate over the volume and use the definition of the leakage probability such that:
Using the divergence theorem one can rewrite the first term :
The relation between leakage and absorption rate is given by:
This illustrates the role of B2i: for a given volume the leakage
probability increases with the mode.
€
Riabs ≡ ψ i
r x ( )
V∫ Σ adV = Σ a ψ ir x ( )dV
V∫
€
∇2
V∫ ψ ir x ( )Σ adV = −Bi
2 ψ ir x ( )∫ Σ adV = −Bi
2Riabs
€
a ∇2ψ ir x ( )
V∫ dV = Σ a
r ∇ ⋅
V∫r
∇ψ ir x ( )dV = −
Σ a
D
r ∇ ⋅
r J i
V∫r x ( )dV
€
=− a
D
r J i
r x ( ) ⋅d
r S
S∫ = −1
Lc2 Ri
leak wherer J i
r x ( ) = −D
r ∇ ⋅ψ i
r x ( )
€
Rileak = Ri
absLc2Bi
2 (3)
Y. Kadi / October 17-28, 2005 43
Leakage Probability
Leakage and non-leakage probabilities:
where ki is the criticality factor for mode i :
given that is a function that
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 even more sub-critical.
A new expression of the flux can be derived as a function of ki :
It is of interest to note the Amplification factor 1/(1-ki) specific to every single mode.
€
Pileak =
Rileak
Rileak + Ri
abs =Lc
2Bi2
1 + Lc2Bi
2 ;
€
Pinon −leak = 1 − Pi
leak =1
1 + Lc2Bi
2 =ki
k∞
€
ki =k∞
1 + Lc2Bi
2
€
Bi2 ≡ Bl,m,n
2 = π 2 l 2
a2 +m2
b2 +n2
c 2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
l,m,n =cl,m,n
Bl,m,n2 − BM
2 =cl,m,n
Bl,m,n2 −
k∞ −1
Lc2
=Lc
2cl,m,n
1 + Bl,m,n2 Lc
2 ×1
1 − kl,m,n
Y. Kadi / October 17-28, 2005 44
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:
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 sub-critical system is therefore keff = k1. This is what is happening when the proton beam is switched off in the energy amplifier.
€
r
x ( ) = Φ l,m,nψ l,m,nr x ( )
l,m,n∑ = Lc
2 cl,m,n
1 + Bl,m,n2 Lc
2 ×ψ l,m,n
r x ( )
1 − kl,m,nl,m,n∑
Y. Kadi / October 17-28, 2005 45
Neutron multiplication in a sub-critical 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 :
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 :
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.
€
M = 1 + k + k 2 + k 3 + ... + k n =k n +1 −1
k −1n→∞ ⏐ → ⏐ ⏐ 1
1 − kfor k < 1
€
k =M −1
M= 1 −
1
M< 1
€
ksrc =M −1
M
Y. Kadi / October 17-28, 2005 46
Calculation of the multiplication factor
By definition the neutron multiplication factor is given by :
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.
By using equation (3) together with the definition of l,m,n we obtain:
hence :
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.
€
M ≡
Φ l,m,nRl,m,nabs + Φ l,m,nRl,m,n
leak
l,m,n∑
Qwhere Q = C
r x ( )dV
V∫
€
M =
Φ l,m,nRl,m,nabs 1 + Lc
2Bl,m,n2
[ ]l,m,n∑
Q=
Lc2Σ ac l,m,n
1 − kl,m,n
ψ l,m,nr x ( )dV
V∫l,m,n∑
Q
€
M = M l,m,nl,m,n∑
Dcl,m,n ψ l,m,nr x ( )dV
V∫Q
and M l,m,n =1
1 − kl,m,n
€
Q = D cl'm'n' ψ l ',m',n'r x ( )
V∫l',m',n'∑ dV
Y. Kadi / October 17-28, 2005 47
Time Dependence
The diffusion equation of neutrons in a non-equilibrium system can be written as:
where v is the mean velocity of the neutrons, so that = 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:
Substituting in (4), following the properties of ψl,m,n lead to :
this implies that the coefficient of every mode must be zero.
€
∂nr x , t( )
∂t=
1
v
∂Φr x , t( )
∂t= D∇2Φ
r x , t( ) + k∞ −1( )Σ aΦ
r x , t( ) + C
r x , t( ) (4)
€
r
x , t( ) = Φ l,m,nl,m,n∑ ψ l,m,n
r x ( ) f l,m,n t( )
€
1
v
df t( )
dt+ DBl,m,n
2 + (1 − k∞ )Σ a[ ] f (t) ⎧ ⎨ ⎩
⎫ ⎬ ⎭l,m,n
∑ Φ l,m,nψ l,m,nr x ( ) = 0
Y. Kadi / October 17-28, 2005 48
Time Dependence
Every mode has therefore its own time dependence, solution of the following equation :
where :
the flux can then be expressed as :
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).
€
df (t)
f (t)= −v DBl,m,n
2 + (1 − k∞ )Σ a[ ]dt
€
f (t) = e−v DBl ,m,n
2 + 1−k∞( )Σ a[ ]t = e−vΣ a 1+Lc
2B l,m,n2
( ) 1−kl,m,n( )t
€
r
x , t( ) = Φ l,m,nl,m,n∑ ψ l,m,n
r x ( )e
−vΣ a 1+Lc2B l,m,n
2( ) 1−kl ,m,n( )t
Y. Kadi / October 17-28, 2005 49
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 :
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:
Let u/r = (r), the equation reduces to :
remember that 1– k > 0.
€
∇2Φ +k∞ −1( )
Lc2 Φ = −
C
D(1)
€
d 2Φ(r)
dr 2 +2
r
dΦ r( )
dr−
1 − k∞
Lc2 Φ r( ) = 0
€
d 2u
dr 2 −κ 2u = 0. where κ ≡1 − k∞
Lc2
Y. Kadi / October 17-28, 2005 50
Neutronic characteristics of a system intrinsically sub-critical
Since 2 is a positive quantity, the general solution is thus:
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:
upon inserting the value for A, it follows that
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 [c = 3 cm in Pb for instance]).
€
u = Ae−κr + Ceκr
€
J (r) = −DdΦ r( )
dr(Fick' s law) and Q = limite
r→04πr 2J r( )( )
€
r( ) =Q
4πDre−κr
€
=Ae−κr
r+ C
eκr
r
Y. Kadi / October 17-28, 2005 51
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 () :
hence :
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)
€
Mc2 ≡ Lc
2 + τ where τ E( ) ≡D
ξΣsEE
E0∫ dE and ξ the average letharg y
€
α =1−k∞
Mc2
Y. Kadi / October 17-28, 2005 52
Spatial distribution of the neutron flux
Y. Kadi / October 17-28, 2005 53
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, SpainCSNSM, 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 54
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 55
The FEAT Assembly
3.6 tonne of natural uranium immersed in water
Y. Kadi / October 17-28, 2005 56
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 57
Simluation of FEAT
Example of a secondary shower produced by a single proton
Y. Kadi / October 17-28, 2005 58
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 Uraniumconverter
Clustering Location
Gas IonisationChamber4 ata Argon
Circular window, ∅=15mmverticalplane
1mg/cm2
deposit1arrayof6,spaced10cmverticall y .2arraysof8,spaced12.3c mvertically
i nwater,betweenUbars
PolycrystallineSiDiodethickness300m
Rectangular.window9.6x 11.6 mm2
vertical plane
1 mg/cm2
deposit10 arrays of 16counters, spaced6.4 cm vertically
in water,between U bars
PolycrystallineSi Diodethickness 300μm
90° sectorr = 14 mmhorizontal plane
1 mg/cm2
deposit4 clusters of 6counters spaced21.3 cm vertically
in fuel bar,between Ucartridges
Thermistancesin metallic probes
U Cylinders : ∅=8 mmPb Cylinder : ∅=10 mm
~ 55 g U 3 thermometers withtwo U probes
in water,between U bars
Lexan foils trackdetectors
Equilateral triangles r =37mmCircles ∅=32 mmRectangle 10x25 mm2 (Vert. Plane)
~ 1 mg/cm25 vertical sets of 2detectors
in water,between U bars
Y. Kadi / October 17-28, 2005 59
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 60
Flux measurement
The neutron flux is parametrized according to an exponential function :(x,y,z) = const. exp(–d/) where the beam is along the x-axis, andd = [(x + )2 +y2 +z2]1/2 where is a constant accounting for the average axial displacement of the spallation shower with the beam energy and is the first moment of the longitudinal distribution of the shower.
Y. Kadi / October 17-28, 2005 61
Determination of the flux parameters
is independent of the beam energy whereas decreases slightly when the beam energy increases.
Y. Kadi / October 17-28, 2005 62
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 63
Measurement of the neutron flux in FEAT
Comparison between different Monte Carlo simulations
Y. Kadi / October 17-28, 2005 64
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 65
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 s /event/CPU)
such a complex tool need to be thoroughly validated (FEAT, TARC, IAEA Benchmarks)
Y. Kadi / October 17-28, 2005 66
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 67
Review of Sub-Critical Core Experiments
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.
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 68
The MUSE Experiment
MASURCA facility (courtesy of CEA)
The Pulsed Neutron Source
« GENEPI »
Y. Kadi / October 17-28, 2005 69
The MUSE Experiment
Y. Kadi / October 17-28, 2005 70
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
Apparent worth of the sourceΦ*
Stochastic
Buffer : Na or SSΦ*
Pulsed
Spectrum sourceDynamic measurementsSpectrum index (test)
Pulsed
Dynamic measurementsM.A. Fission ratesNeutron spectrometryLead zone
Y. Kadi / October 17-28, 2005 71
General view of the YALINA fuel subassembly.
The YALINA Experiment
Y. Kadi / October 17-28, 2005 72
The YELINA Experiment (2)
Y. Kadi / October 17-28, 2005 73
The YELINA Experiment (4)
Experiment & calculations:
keff vs. fuel load
Y. Kadi / October 17-28, 2005 74
Neutron pulses measured by 3He-counters in different experimental channels.
Layout of the Yalina core
–Close to criticality: straight line, α constant
–Deep subcriticality: α time dependent
Main YALINA Results
Y. Kadi / October 17-28, 2005 75
Physics Validation Sequence
CONFIG SOURCE KINETICS POWER EFFECTS
FEAT SPALL THERMAL NO
MUSE DD/DT FAST NO
YALINA DD/DT THERMAL NO
YALINA DD/DT FAST NO
TRADE SPALL THERMAL YES
RACE -NUCL THERMAL NO
SAD SPALL FAST NO
EA SPALL FAST YES
CONFIG SOURCE KINETICS POWER EFFECTS
FEAT SPALL THERMAL NO
MUSE DD/DT FAST NO
YALINA DD/DT THERMAL NO
YALINA DD/DT FAST NO
TRADE SPALL THERMAL YES
RACE -NUCL THERMAL NO
SAD SPALL FAST NO
EA SPALL FAST YES