“Neutronics” · Scattering • The particle is deviated • The target nucleus: – Does not...

Introduction.. 1 Neutronics

Laboratory for Reactor Physics and Systems Behaviour

Introduction, Brief Review of Nuclear Physics

R. Chawla

“Neutronics”

Introduction.. 2


Neutronics

Neutronics: Physics of Neutrons, Reactor Physics

“Bridge” between nuclear physics and nuclear reactor design

Deals with the behaviour (interaction, transport) of neutrons in matter • Allows one to study the neutron

balance in a nuclear reactor

Forms the basis for the production of nuclear energy from fission • Clearly, for reactor design, other

disciplines also crucial: thermal-hydraulics, materials,

chemistry, instrumentation & control, radioprotection, ..

Cross-sections(different reactions)

Neutron behaviour(balance)

Thermal-hydraulics,Materials, etc.

Reactor Design

Neutronics

Nuclear Physics

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Neutronics

Linkage to Other Energy-Related Courses (Physics-Master)

Reactor Technology • 2nd half of semester, same time-table as “Neutronics”

– Heat removal phenomena in a reactor core – Reactor design – Light water reactors (PWRs, BWRs) – Other types of nuclear power plants – Generation IV systems

Advanced Fossil and Renewable Energy Systems • Conventional and novel non-nuclear energy production, regular 14-week course

– Thermodynamic cycles (energy conversion) – Steam power plants, gas turbines, combined cycles, heat pumps – Direct energy conversion (fuel cells) – Hydroelectricity, solar energy, wind power, biomass,… – Environmental effects of energy production

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Neutronics

General organisational aspects

Each Wednesday, both a.m. and p.m.: 2x45' lectures, 1x45' exercises

Exercise period: numerical examples, discussion of “homework”

Oral exam in January 2009, for award of 4 credits • 30' : 1 general question + 1 practical question or exercise (15' each) • 30' : for preparation, with documentation

Course material: text books, special references • Elements of Nuclear Engineering, J. Ligou (1986; chapters to be distributed)

– Effectively, English translation of “Introduction au génie nucléaire” (PPUR, 1997) • Introduction to Nuclear Reactor Theory, Reprint, J. R. Lamarsh (ANS, 2002) • Elementary Reactor Physics, P. J. Grant (Pergamon, 1966) • Nuclear Reactor Physics”, 2nd ed., W. M. Stacey (John Wiley, 2007)

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Neutronics

Course Contents - 1

Lesson 1: Introduction, nuclear physics basics • Atom, nucleus, radioactivity, nuclear reactions

Lesson 2: Fission • Reaction characteristics, nuclear fuels, neutron cross-sections

Lesson 3: Neutron spectra, thermal cross-sections, simple neutron balance

Lesson 4: Classification of reactors according to power, neutron propagation

Lesson 5: Angular flux, neutron current, neutron balance equation, Fick’s law

Lesson 6: Transport equation, diffusion equation as special case • Typical diffusion theory solutions

Lesson 7: Slowing down, moderator characteristics, slowing down equations

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Neutronics

Course Contents - 2

Lesson 8: Slowing down spectra, resonance escape probability, Fermi age

Lesson 9: Multiplying media (reactors) • Bare-reactor equation, bucking, flux distributions

Lesson 10: Modified one-group theory, migration area, thermal/fast reactors

Lesson 11: Multi-zone reactors, reflectors (1-, 2-group) • Multi-zone, multi-group generalisations

Lesson 12: Heterogeneous reactors, reactor kinetics without/with delayed n’s

Lesson 13: In-hour equation, stable period, prompt jump, control rod calibration

Lesson 14: Reactivity variations (short-, medium-, long-term) • Reactivity coefficients, xenon, fuel burnup (composition changes)

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Neutronics

Lesson 1: Nuclear Physics Basics, Reactions, Radioactivity

Atoms and Nuclei

Chart of the Nuclides

Binding Energy, Nuclear Reactions

Radioactivity, Radioisotopes

Beam Intensity (Flux), Types of Interactions

Cross-sections (Microscopic, Macroscopic)

Fission/Fusion Differences

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Neutronics

Atomic and Nuclear Structure

Rutherford’s model of the atom • Mass concentrated in the nucleus (mH/me ~ 1837) • Nuclear charge: +Ze (Z: atomic number, e ~ 1.6.10-19 coulomb) • Quantum mechanical basis for atomic, nuclear structure • “Classical dimensions”: nucleus ~ 10-13 cm, atom ~ 10-8 cm

Energy units (1eV ~ 1.6.10-19 J) • Binding energy of outermost electrons ~ order of eV (energy involved in chem. reactions) • Binding energy of nucleons (constituents of nucleus) ~ order of MeV !

Constituents of nucleus: protons, neutrons • Particles of very similar mass, proton charged (+e), neutron neutral • In terms of elementary particles, both are hadrons, made of quarks (bound by gluons);

electrons are leptons (as are positrons, neutrinos) • Nucleus: Z protons (atom neutral), A-Z neutrons (A: atomic mass)

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Neutronics

Chart of the Nuclides Stability of nucleus depends on N/Z

• For light stable atoms, N~Z • For Z>20, N>Z: strongly attractive force betn.

nucleons compensates repulsive coulombian force betn. protons

Unstable nuclei, radioactive (natural, artificial) • ZXA → Z-2YA-4 + 2He4 (α-decay) .. heavy • ZXA → Z+1YA + e- + νo (β--decay) .. n-rich • ZXA → Z-1YA + e+ + νo (β+-decay) .. n-poor • ZXA + e- → Z-1YA (electron capture) .. n-poor • ZXA* → ZXA + γ (γ-decay) .. excited

Nuclei of same Z with different N: isotopes • e.g. 1H1, 1H2, 1H3

Nuclei of same mass: isobars • e.g. 53I139 → 54Xe139 → 55Cs139 → … (β--decay chain)

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Neutronics

Atomic Mass, Mass Defect, Binding Energy

Atomic mass A (A gm → NA atoms) • NA (Avogadro’s No.) ≈ 6.023. 1023 • 1 amu = 1/12 mass of C12 atom ≈ 1.66. 10-24 gm • n, p masses ≈ 1.0087, 1.0073 amu

Mass of nucleus < sum of nucleon masses

Mass defect: ∆m = Zmp + (A-Z)mn - mX

Binding energy Eb = ∆m.c2 (Einstein)

Eb/A, measure of force betn. nucleons

Sharp increase at low A value, broad maximum at ~ A=50 Binding energy / nucleon

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Neutronics

Nuclear Reactions, Reaction Energy

Radioactivity, particular example of a nuclear reaction • Single reactant (cf. chemical dissociation)

In general, X1 + X2 → X3 + X4 , e.g. • 2He4 + 4Be9

→ 6C12 + 0n1 or 4Be9 (α,n) 6C12 … discovery of neutron

Energy of reaction : Q = (Eb)3 + (Eb)4 - (Eb)1 - (Eb)2

= (∆m.c2)3 + (∆m.c2)4 - (∆m.c2)1 - (∆m.c2)2

= (m1 + m2 - m3 - m4).c2

Energy / mass equivalence : 1 amu ≈ 1.66.10-24 g x (3.1010 cm/s)2 = 1.492.10-3 erg = 931 MeV

Q = (m1 + m2 - m3 - m4).c2 = 931.(m1 + m2 - m3- m4) MeV

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Energy-Releasing Reactions Reactions which result in a shift towards

the broad maximum • Eb , ∆m increase (products more stable) • Energy released (reaction: exoenergetic)

Two possibilities: • Light nuclei fuse: movement from left

towards maximumum.. fusion • Heavy nucleus splits in two: movement

from right to maximum.. fission

Example : (d,d) fusion reaction..

1H2 + 1H2 → 1H3 + 1H1

Q = 931.(2.0141 + 2.0141 – 3.0166 – 1.0073) ≈ 4.0 MeV

Fission of a heavy nucleus.. 92U235 + 0n1

→ 2 F.P. + (2 to 3) 0n1

Binding energy / nucleon

fission

fusion

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Neutronics

Radioactivity Calculations

Often encountered in NE.. e.g. fuel, activation, fission products, wastes

Fundamental law for a radionuclide (radioisotope):

Units of (radio)activity: • Historical.. 1 curie (Ci) = 3.7 x 1010 dis/s (activity of 1 g of Ra226) • Actual.. 1 becquerel (Bq) = 1 dis/s • For example: 1 mCi = 10-3 Ci = 3.7 x 107 Bq = 37 MBq

By integration: tt eAtAeNtN λλ −− == .)0()(,.)0()(

€

dN(t)dt

= −λN(t) = A(t)

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Half-Life, Mean Life

Half-Life : time for N(t) or A(t) to become half initial value

Mean Life : average life, i.e. weighted with the corresponding number of nuclei

21T

€

N(T1 2)N(0)

= e−λ t =12

⇒ T1 2 =ln2λ

≅ 0.693λ

€

t

Thus,

€

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Neutronics

Radioactive Equilibrium One often finds radioactive decay chains, e.g.

• 92U238 → 90Th234 → 91Pa234 → 92U234 → … (U238 series: successive α-, β--decays)

• Fission product β--decay chains (earlier example: isobars)

• In general, X1 → X2

→ X3 → …

• Complex evolution, but sometimes simple case of » , , … One then has, after a certain time : λ1 N1 ≈ λ2 N2 ≈ λ3 N3 ≈ … ≈ constant

With the derivatives ≈ 0, for isotope i (secular equilbrium)

€

dN1(t)dt

= − λ1N1(t)

€

dN3(t)dt

= λ2N2(t) −λ3 N3(t)

€

dN2(t)dt

= λ1N1(t) −λ2 N2(t)

(precursor)

…etc. (descendents)

( )121

T ( )221T ( )

321T

ii

NNλλ 11≈

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Neutronics

Use of Nuclides Chart Z – N diagram representing all possible nuclides : stable / radioactive, natural / artificial

Various references, e.g. http://www.nndc.bnl.gov/chart/

Can deliver detailed information on : half-life, decay scheme, emiitted radiation type / energies, etc.

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Radioactivity Induced by Neutrons (Activation)

Neutrons can “activate” materials easily • No “resistance” of electrostatic field of nucleus • Neutron capture, i.e. (n,γ) reactions, produce n-rich radioisotopes • Absorption probability (cross-section) higher at

low (“thermal”) neutron energies

Example: 27Co59 + 0n1 → 27Co60 + γ ( ≈ 5.3 y )

For a production rate S per sec,

Integrating,

or

For t0 much greater than :

For t0 much smaller than :

After t0 :

21T

)()( tNSdttdN

λ−=

€

N(t) =Sλ1 −( e−λt )

( )teStA λ−−= 1)(

21T

21T

€

A(t0) ≅ λt0S

€

A(t0) = S

€

A(t) = A(t0) e−λ( t− t0 )

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Flux of Particles, Interaction Rate

Fission, fusion are exoenergetic • What is their probability of occurrence?

Monoenergetic particle beam & a target • Density of particles in beam = n (cm-3) • Intensity (flux, cm-2 s-1), I = n v (v : velocity, cm s-1)

Total interaction rate with nuclei in target R ∝ I N V = σI N V (V : volume of target, cm3) σ : cross-section, probability of interaction

• Depends on type, particle energy • Pro target nucleus, r = σI (σ : microscopic c-s)

Pro cm3 of target, R = σ NI = ΣI (Σ : macroscopic cross-section)

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Cross-sections: Dimensions, Units

σ : dimensions of an area (cm2) • r (per nucleus, s-1) = σ (cm2) . I (cm-2 s-1) • “Effective area” offered by the nucleus for the interaction-type involved • Unit : 1 barn (b) = 10-24 cm2

Values vary ∼ from hundreds of barns to a few millibarns (mb)

For Σ (σN), dimensions: cm-1

• R (cm-3 s-1) = Σ (cm-1) . I (cm-2 s-1)

Σ : effectively the probability of interaction as particle traverses 1 cm of target

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Types of Interactions

Scattering • The particle is deviated • The target nucleus:

– Does not change (elastic scattering)) – Is excited (inelastic scattering)

Absorption • The particle is absorbed by the nucleus, the products are new, e.g.

– Radiative capture: ZXA + 0n1 → ZXA+1 + γ

– Fission, a special case: 92U235 + 0n1

→ 2 F.P. + (2 to 3) 0n1

– Other types (less important): (n,2n), (n,3n), (n,α),…

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Neutronics

Formation of a compound nucleus

Neutron of energy En absorbed in ZXA to form compound nucleus

ZXA+1

• Excitation level ∼ En + L (B.E. of last nucleon)

Several different results possible, e.g. • Neutron capture: nucleus decays to ground state,

via emission of capture γ’s • Inelastic scattering: neutron re-emitted, with residual

(ZXA) nucleus in excited state (inelastic γ emitted) • Compound elastic scattering: residual nucleus

returned to ground state (Elastic scattering usually without CN formation…

potential scattering)

CN formation probability high if ZXA+1 has excitation state near En + L (quantum mechanics) • cross-section resonance at En

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Cross-section Notations

Scattering: σs σs = σe + σi (elastic, inelastic)

Absorption: σa σa = σf + σc (fission, capture)

Total cross-section: σt σt = σs + σa = σs+ σf + σc

Macroscopic cross-sections : Σt = Nσt , Σa = Nσa , Σf = Nσf , etc.

For a mixture of nuclei: , etc.

€

N j σ t( ) j[ ]j∑

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Neutronics

σ, Σ… Functions of Energie, e.g.

σ (U235) ↑ as neutron energy ↓

• No resistance from electrostatic field of the nucleus

Neutrons slowed down in a reactor (use of a moderator)

Lowest energy possible: n’s in thermal equilibrium with moderator atoms:

Eth ∼ 0.0235 eV at 20°C

⇒ σf ∼ 600 b!

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Fission, Fusion Differences

For fusion reactions e.g. (d,t): 1H2 + 1H3 → 2He4 + 0n1 (d,d): 1H2 + 1H2 → 1H3 + 1H1

• σ = 0 for E < Es (Es : threshold ∼ 10 keV)

• The particles need to overcome the “coulombian” barrier (have energy > Eth)

Scattering, a big help in fission (slowing down), great disadvantage in fusion

Solution: have a thermal equilibrium with Eth > Es … (∼ 10 keV → 108 K !)

• The ionised medium needs to be heated tremendously (plasma)

→ Thermonuclear fusion… still a great technological challenge!

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Summary, Lesson 1

Nucleus: protons + neutrons; former: atomic number Z; the two together: atomic mass A

Heavy nuclei: richer in neutrons

Energy in a nuclear reaction: linked to binding energies (mass defects) of reactants • Fission, fusion: “movement” towards the large maximum of the BE-curve

Radioactivity, specific type of nuclear reaction (natural & artificial) • The disintegration rate, or activity (becquerels), given by… • Neutrons in a reactor “activate” materials (production of radioisotopes)

Reaction rate = Flux x Cross-section (microscopic, macroscopic)

Different types of reactions: absorption (fission, capture,…), scattering (elastic, inel.),…

Energy dependence of cross-sections (e.g. fission, fusion)

€

A(t) =dN(t)dt

= −λN(t)

“Neutronics” · Scattering • The particle is deviated • The target nucleus: – Does not...

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Transcript of “Neutronics” · Scattering • The particle is deviated • The target nucleus: – Does not...