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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. LECTURES OUTLINE. LECTURE 1: Physics of Spallation and Sub-critical Cores: Fundamentals

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iaea ictp workshop on technology and applications of accelerator driven systems ads

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

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)
physics of spallation sub critical cores fundamentals

Physics of Spallation & Sub-critical Cores: Fundamentals

Y. Kadi

CERN, Switzerland

17 October 2005, ICTP, Trieste, Italy

introduction to ads
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 a-particles (14N7 + 4He217O8 + 1p1)

 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

introduction to ads1
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.

historical background
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).

the energy amplifier
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.

conceptual study of an energy amplifier
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)
review of existing ads concepts
Review of Existing ADS Concepts

p Classification of existing ADS concepts according to their physical features and final objectives:


 neutron energy spectrum

fuel form (solid/liquid)

fuel cycle

coolant-moderator type

final objectives

the spallation process 1
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

properties of the spallation induced secondary shower
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.
the spallation process 2
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 so-called ADS

the spallation process 3
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...

the spallation process 4
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).

spallation neutron yield
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.

spallation neutron spectrum
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.

spallation product distribution
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

energy deposition
Energy Deposition

pExample 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.

models and codes for high energy nuclear reactions fluka
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

fluka description
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
fluka hadronic models
FLUKA – Hadronic Models

Inelastic 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 /g-deexcitation of the residual nucleus

Elastic Scattering

  • Parameterized nucleon-nucleon cross sections.
  • Tabulated nucleon-nucleus cross sections
hadron nucleon cross section
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

inelastic hn interactions
Inelastic hN interactions

Intermediate Energies

  • N1 + N2 N1’ + N2’ + pthreshold around 290 MeV important above 700 MeV
  • p + N  p’ + p” + N’ 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

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

ha at high energies
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
generalized intranuclear cascade
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 probabilitysfree + 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
advantages and limitations of ginc

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


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)

Advantages and Limitations of GINC
residual nuclei
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)
electrons and photons in fluka
Electrons and Photons in FLUKA


  • Electro Magnetic FLUKA (EMF) at a glance
  • Physical Interactions
  • Transport
  • Biasing
low energy neutron transport in fluka
Low Energy Neutron Transport in FLUKA


  • 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
sub critical systems
Sub-Critical Systems
  • In Accelerator-Driven Systems a Sub-Critical blanket surrounding the spallation target is used to multiply the spallation neutrons.
sub critical vs critical systems
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...

reactivity insertions
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 $ (Dk/k ~ 6.510–3) 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 $ = r/b; r = (k–1)/k) :

physics of sub critical systems
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]; F = neutron flux [n/cm2/s]

n = neutrons emitted per fission; Sf = macroscopic fission cross section

= external neutron source (spallation neutrons for instance)

Sa = macroscopic absorption cross section(capture + fission)

= neutron current [n/cm2] according to Fick’s Law


for (n,xn)

ext. source


physics of sub critical systems1
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, F et C are only functions of the space variables
    • where k is defined as: (n,xn included in n)
physics of sub critical systems2
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)
material and geometric bucklings
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"(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.
material and geometric bucklings1
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:
    • 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:
    • DB2M/Sa represents the excess multiplication which is necessary to compensate for the leakage of neutrons
typical solutions
Typical solutions
  • The geometry of the system determines the type of oscillatory solution. In a critical system, the fundamental mode alone is important.
study of a simplified sub critical system
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 :

study of a simplified sub critical system1
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 :
  • Introducing these expressions in the equation one can determine the coefficients Fl,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 B2l,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).
leakage probability
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 Sa, 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 absorptionrate is given by:This illustrates the role of B2i: for a given volume the leakage probability increases with the mode.

leakage probability1
Leakage Probability
  • Leakage and non-leakageprobabilities: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.

flux in a sub critical system with k 1
Flux in a Sub-Critical Systemwith k > 1
  • The general solution of the flux for a finite system with k > 1 is given by:
  • Clearly k1 is higher than knfor 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.
neutron multiplication in a sub critical system
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 ksrcthe value of k calculated from the net multiplication factor M in the presence of an external source.
calculation of the multiplication factor
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 Fl,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.
time dependence
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 F = nv.
  • Consider the case of a neutron burst represented by C0d(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 yl,m,n lead to :this implies that the coefficient of every mode must be zero.
time dependence1
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).
neutronic characteristics of a system intrinsically sub critical
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 = F(r), the equation reduces to :remember that 1– k > 0.
neutronic characteristics of a system intrinsically sub critical1
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 k 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]).
particular case of feat
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) :

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)
the feat f irst e nergy a mplifier t est experiment 1993 94
The FEAT (First Energy Amplifier Test) Experiment 1993-94


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

  • Top and side view of the FEAT assembly, on the T7 beam line of the CERN-PS complex.
the feat assembly
The FEAT Assembly
  • 3.6 tonne of natural uranium immersed in water
properties of the spallation induced secondary shower1
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.
simluation of feat
Simluation of FEAT
  • Example of a secondary shower produced by a single proton
detectors used in feat
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).
determination of the multiplication factor k
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
flux measurement
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, andd = [(ax + d)2 +y2 +z2]1/2 where d 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.
determination of the flux parameters
Determination of the flux parameters
  • l is independent of the beam energy whereas a decreases slightly when the beam energy increases.
flux measurement1
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.
measurement of the neutron flux in feat
Measurement of the neutron flux in FEAT
  • Comparison between different Monte Carlo simulations
measurement of the energy gain
Measurement of the Energy Gain
  • Behaviour of the measured gain matches MC simulation  FEAT good benchmark for validating calculation methods and nuclear data.
  • 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)
conclusions from feat
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.
review of sub critical core experiments
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.

the muse experiment
The MUSE Experiment

MASURCA facility (courtesy of CEA)

The Pulsed Neutron Source



The YALINA Experiment

General view of the YALINA fuel subassembly.

the yelina experiment 4
The YELINA Experiment (4)

Experiment & calculations:keff vs. fuel load

main yalina results
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

Layout of the Yalina core

physics validation sequence
Physics Validation Sequence