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p , τ .. 1

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Lesson 8: Slowing Down Spectra, p , Fermi Age

Slowing Down Spectra in Infinite Homogeneous Media

Resonance Escape Probability ( p )

Resonance Integral ( I , Ieff )

p , for a Reactor Lattice

Semi-empirical Relations for Ieff

Neutron Migration during Slowing Down

Fermi Age Theory

Physical Significance of Age ( τ )

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Slowing-Down Energy Spectra in Infinite, Homog. Media OK FROM HERE !!! In a reactor, there are sufficiently large, individual zones..

For an infinite, homogeneous medium … Angular fluxes: isotropic, Scalar flux: uniform, Net current: zero ( )

Eq. (2)… Neutron balance eq. for band E, E+dE …

with q(E) to be obtained from Eq. (1)… Slowing-down source eq. …

⇒ One may consider 2 different cases: I. Σa ≈ 0 (Non-absorbing medium, e.g. moderator…) II. Σa finite (Fuel / moderator mixture…)

(for all )

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Non-absorbing Medium

Very simple neutron balance equation: • In general, Q is the fission-source density (fission spectrum)

Integrating,

For

For (total fission source)

⇒ In absence of absorption (Σa = 0) and of leakage ( = 0), no. of n’s crossing each energy = Qf

for all E < Es (constant slowing-down source)

- No accumulation of n’s at energy E

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Non-absorbing Medium (contd.)

One can show that the solution , with C a constant, gives:

q(E) = constant , for E < Es

Substituting for Φ in

Thus,

Since Σs ~ constant in practice, the slowing-down spectrum is ~ 1/ E (Fermi)

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Comment (1) For a mixture of isotopes, one needs to define such that

If one assumes and Σsi constant,

Thus, , i.e. (as given before)

€

ξ

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Comment (2)

Derivation of the result was done in an approximate manner

In general, for a source of n’s of energy E0 (e.g. ~ 2 MeV, on average, for fission n’s) • The result is the asymptotic solution (for E << E0)

• There are transitions near E0 (at αE0 , α2E0 , …) – E.g., for the first collision, αE0 ≤ E ≤ E0 , etc. – For H1, there are no transitions

• Detailed treatment: Ligou, Section 8.3.2

⇒ Solution of Placzek :

• In practice, the transitions are not very important

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Absorbing Medium

For a fuel-moderator mixture, a few assumptions need to be made:

For E < Es ,

Integrating,

Thus,

even though q(E) ≠ constant

in absence of absorptions

Considering Σa ≈ 0 for E ≥ Es ,

probability of escaping absorption during slowing down from Es to E

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Absorbing Medium (contd.)

With Σs >> Σa , one may take Σs ~ Σt

In spite of the assumptions made, Eq. (3) is valid in certain, very different situations:

• In hydrogeneous media (slowing down in hydrogen, even with Σa > Σs )

• In the region of sharp (narrow), isolated resonances (representative of epithermal absorptions in the fertile isotopes, U238, Th232)

Most important contribution to absorptions during slowing down

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Resonance Escape Probability , p

For a reactor, one has a mixture (moderator, fuel, structure,…)

Reasonable approximation: epithermal absorptions only in fertile material • For others, resonances are generally much less important than thermal absorptions • For fissiles, resonances “compensate” partly (in terms of productions, absorptions)

For p , reference energy is E = Et*

• Energy < first resonance , but > Eth

• Limit rather arbitrary, e.g. ~ 4 eV

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p (contd.)

In , one sets

Thus,

with

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Dilution Cross-section, I , Ieff We have introduced in expression for p :

• dilution cross-section

Ieff : effective resonance integral (depends only on σe ) In the limit σe → ∞ (Nm >> Nc) , I : infinite-dilution resonance integral

(σe depends on the “dilution”, Nm/ Nc …)

In practice, Ieff << I , since σe not that large • One has the phenomenon of self-shielding • Flux depressed within the resonances

For Nm ↑ (σe ↑), self-shielding effect reduced

(max. value)

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p , for a Homogeneous Mixture

We have:

For Nm/ Nc ↑ , Ieff ↑ , but the denominator ↑ more strongly

Effectively, p → 1 for Nm → ∞ , but slowly…

An increase in the slowing-down power

allows neutrons to “jump over” the traps (greater probability of having an energy loss >> width of the resonance)

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p , for a Reactor Lattice In practice, fuel rods regularly spaced in the moderator: heterogeneous lattice Equivalence theorem: where

• considered with respect to the volume of the core • Ieff defined as before, but with where

– characterises fuel-rod dimensions (diam. if cylindrical, otherwise)

– fuel density in the usual sense (i.e. per unit volume of the fuel) – factor characteristic of the lattice (Bell factor)

σe , hence Ieff , independent of moderator • For a given ratio , p ↑ when ↑ (σe → 0) • For → 0 (thin rods), σe → ∞ , pe→ min. (infinite dilution : Ieff max.) • Thus, heterogeneity of a lattice is needed, not only for technological reasons…

€

N c /N m

€

N c /N m

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Semi-empirical Relations for Ieff

In general, with

Experimental measurements of Ieff , for different lattices, yield semi-empirical of the form, e.g.

• Often it is which gives a better “fit”)

Qualitaively, (2) corresponds to resonance absorptions of the type:

Thus,

n’s absorbed in entire volume (σa moderate)

n’s absorbed on rod surface (σa very high)

Per nucleus,

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Neutron Migration during Slowing Down

Till now: infinite, homogeneous media → Φ uniform (same for all )

In practice, one has a reactor of finite dimensions, non-homogeneous • There is a relationship between Φ(E) and distance from the source • Numerical approach (multigroup theory) allows treating n’s in many energy groups

– One can then speak of, for example, a “diffusion area” for each group…

A simplified treatment allows one to obtain analytical solutions (Fermi’s theory)

Corresponding hypotheses: • λt does not vary strongly with energy • ξ is small (slowing down almost continuous) • Σa ~ 0 • Neutron spectrum not affected by differential leakage (greater leakage for fast n’s) • Diffusion theory valid

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Fermi Age Theory One considers the neutron balance in

The change is due to leakage…

In absence of absorptions,

Thus,

Defining “Fermi Age” corresponding to energy E by , i.e.

= ⇒

(Age Equation)

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Solution of Age Equation

The form is that of the time-dependent, heat conduction equation but has the dimensions of area, not of time…

One may use the the method of Fourier integrals to solve the Age Equation • E.g. for a point source in an infinite medium :

• The distribution is Gaussian – For τ = 0 (E = E0) : δ-function at r = 0 – For large τ : flat distribution

• Neutrons of large τ are scattered over large distances, distributed in a uniform manner

• For E = Eth , one obtains the distribution of the thermal source (can be used in combination with the diffusion kernel for thermal n’s)

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Physical Significance of τ

As for L , one may consider the average square of the distance travelled by a neutron for acquiring the age τ

No. arriving with age τ in the shell between r , r+dr

i.e.

For E = Eth , τ = τth … : slowing down length

Thus,

Age is proportional to the average squared distance travelled by a n between emission and its arrival at the corresponding energy E

(important for calculating the leakage during slowing down)

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Summary, Lesson 8

Slowing Down Spectrum in Infinite Non-absorbing Medium

Consideration of Absorption during Slowing Down

Resonance Escape Probability p and Effective Resonance Integral Ieff

Semi-empirical Relations for Reactor Lattices

Neutron Migration during Slowing Down

Fermi Age Equation

Solution for a Point Source

Physical Significance of Age ( τ )