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