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Neutronics for critical fission reactors and sub-critical fission in hybrids Massimo Salvatores (CEA, Cadarache, France). WORKSHOP ON FUSION FOR NEUTRONS AND SUB-CRITICAL NUCLEAR FISSION FUNFI Villa Monastero, Varenna, Italy September 12 - 15, 2011. Outline.

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Neutronics for critical fission reactors and sub-critical fission in hybrids

Massimo Salvatores (CEA, Cadarache, France)





Villa Monastero, Varenna, Italy

September 12 - 15, 2011



  • Hybrids (sub-critical) reactors have been proposed as a tool to help the management of highly radioactive waste resulting from the operation of standard nuclear fission power plants.
  • In fact, neutrons can be used to « transmute » long lived radioactive nuclei into much more short-lived nuclei.
  • In order to understand the potential role of hybrids, it is necessary to clarify which are the physics features that allow them to fulfill that role
  • For this purpose, some basic notions of fission reactor physics will be shortly summarized
  • Successively, the notion of « neutron balance » and « neutron surplus » will be introduced.
  • By these notions it is possible to identify the most adapted nuclear systems to perform transmutation.

Fission and other neutron-nuclei interaction reactions


(A+1) +γ: (n, γ) capture reaction


(n+nucleus with A nucleons)= (A+1)excited

A+n+γ: (n,n‘) inelastic scattering

Reaction channels

(A-1) +2n: (n,2n) scattering



Units and some orders of magnitude

The radius of a nucleus is given by:


where r0=1.2x10-15m and A is the nucleon number.

The following units are generally used:

The scattering cross section for the interaction of a neutron with a nucleus considered as as

spherical target, would be:

σ ~ 4πR2

For a nucleus with A~200, σ~10barn.


Multiplication factor and delayed neutrons

  • Neutrons issued from a fission can successively give a new fission.
  • The number of neutrons produced in average per fission : ~2.5 (e.g. for U- 235)
  • The kparameter is the multiplication factor defined as the ratio of « useful » neutrons produced in average per fission of one generation to the number of « useful » neutrons of the previous generation:

k is the factor by which the neutron population is multiplied going from one generation to the next


The time interval  between two generations is the mean lifetime of neutrons in the medium, i.e. the time interval between their birth by fission and their desappearence by absorption or leakage out of the system.

Its order of magnitude is 10-5-10-7 sec

The evolution of the neutron population n(t) is given by:

or :

then :

and :


The power from the fission is proportional to the fission reaction rate, i.e. proportional to n, according to the same exponential behaviour:

P(t) = P(0).exp[(k - 1) t/]. (P in Watt)

If t = 1 s,  = 2.5x10-5 s and, respectively, k = 1.0001 and k = 0.9999 we could expect:

P(1) = 55 P(0) and P(1) = 0,018 P(0)

However, things are different due to the presence of a fraction of neutrons that are emitted at fission with some « delay » (delayed neutrons). For example in the case of U-235, 99.32 % of the neutrons are « prompt » and 0.68 % are « delayed ».

These delayed neutrons come from the radioactive decay of some fission products, with periods of ~0.2 sec- 1 min.

And this is enough to change the kinetic behavior of a reactor!


If  is the fraction of delayed neutrons,  their average emission delay, and the mean neutron lifetime, as before

now the average generation time separating two fissions is equal to :

= (1 - ) x  +  x ( + )=  + .

In the case of U-235:

= 2.5 x 10-5 + 0.00679 x 11.31 = 0.077 s

The second term is dominating, and delayed neutrons change completely the overall generation time.

In the previous examples, the power increase/second is ~0.1% (and not a factor 55!) and similarly the power decrease is ~(-0.13%) and not a factor ~50.


Neutron Transport Equation

The most general neutron transport equation (Boltzmann equation) is a balance statement that conserves neutrons. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It is formulated as follows:


Thermal or fast neutrons?

  • The most fundamental technological difference between nuclear fission reactors concerns the means by which the problem of sustaining a chain reaction is achieved.
  • One solution is to slow down neutrons to so-called "thermal" energies (around 0.025 eV) by using a « moderator ». This has the advantage of allowing a chain reaction to be sustained using natural or slightly enriched uranium, and almost all of the world's operating power reactors employ this solution -these are known as "thermal reactors". If water is the moderator, we speak of « Light Water Reactors », LWR
  • The disadvantage of this approach is that only 0.7%of uranium produces useful energy. This can be overcome by increasing the proportion of fissile atoms by enrichment, or by using plutonium, and by constructing the reactor without a moderator. In this case the average energy of the neutrons in the core is much greater than in thermal reactors (they are known as "fast" neutrons).
  • In this case there is an excess of neutrons that can be used:
    • to convert the non-fissile U-238 isotope (99.3% of uranium) into Pu-239, which is fissile
    • and/or to « transmute » specific elements (e.g. nuclear wastes)

Neutron Moderation and Cross Sections Comparison

  • Significant elastic scattering of the neutrons in both spectra
  • However, in FRs neutron moderation is much less since high A materials are used
  • If sodium is chosen as coolant in FR, it is also the most moderating material
  • In LWRs, neutrons are moderated primarily by hydrogen
  • Slowing-down power in FR is ~1% that observed for typical LWR:

Minimum energy of a neutron after elastic collision is determined by the parameter α: Emin= αE where:

  • Thus, fast neutrons are either absorbed or leak from the reactor before they can reach thermal energies

The fission/absorption ratios are consistently higher for the fast spectrum SFR. Thus, in a fast spectrum, actinides are preferentially fissioned, not transmuted into higher actinides


How cross section characteristics translate into neutron balance: the concept of neutron consumption/fission.

  • Consider an infinitely large homogeneous core that is fed by several actinides with a given rate S (nuclides/s) and a continuous discharge of part of its fuel inventory for reprocessing and/or storage in a repository.
  • The processes of transmutation of the incoming nuclides (the “fathers”) under neutron flux gives rise to “families”.
  • The father and his family are producing neutrons up to their “death” (complete disappearance).
  • The reasons for the nuclide destruction can be:
  • nuclide fission,
  • natural decay,
  • fuel cycle procedures, either discharge to storage or fuel losses during reprocessing.
  • It is possible to calculate how many neutrons each father (together with his family) is able to consume/produce during its irradiation.

The total number of neutrons Dj consumed by the given J-family can be calculated according to the following scheme:

where PJNr J(N +1)s is the probability of transmutation of the nuclide JJNr (belonging to

the Nth generation within family J) into nuclide J(N +1)s (belonging to the (N+1)th

Generation within family J). All these nuclides are the members of the J-family.

is the number of neutrons consumed during transition AB and depends on the nuclear reaction type

  • The neutron consumption values for each reaction type are defined as follows:
  • Neutron capture (n,): R=1,
  • (n,mn) reaction: R=1-m
  • Fission: R=1-
  • Natural decay: R=0

Positive D means “consumption” and negative D means “production”.


In practice, starting with the “father” isotope in family J, one evaluates the probability of each nuclear reaction, calculating at the same time the number of neutrons consumed as a result of each reaction.

This procedure is systematically repeated up to (almost) complete disappearance of the J- family.

The following figure presents the U-235 chain (family structure) as a simplified example:






Neutron balance, neutron surplus and transmutation

  • The linearity of the nuclide concentration equation with respect to the fuel feed allows a very simple algorithm of the D-evaluation for cores loaded with different “father” nuclides simultaneously.
  • The total neutron consumption of the core (Dfuel) is
  • where J is the fraction of J-family in the feed stream.
  • The global neutron balance of a core must consider the total neutron production (consumption) of the fuel, the parasitic captures of other core components (Cpar) and of the accumulated fission products (CFP), and neutron leakage (Lcore).
  • The general equation for the Neutron Surplus (NScore) then becomes:
  • Dfuel - Cpar - CFP – Lcore = NScore
  • To have a critical system, the neutron surplus should be ≥0

NScore values for a typical Light Water (Thermal) Reactors, LWRs and Sodium Cooled Fast Reactors, SFRs


As for « transmutation », several features characterize the transmutation potential of a specific neutron field for each isotope, e.g.:

Fission of isotope A should be favoured against (n,γ) and (n,xn) reactions. I.e., starting from isotope A reactions giving rise to A+1, A+2 etc should be minimized.

The isotopes that successively lead towards full fission should, as much as possible, be “neutron producers” rather than “neutron consumers”, in order to allow a viable core neutron balance and surplus

As for point 1, we have seen that the fission to absorption ratio is more favorable (i.e. larger) in a fast neutron system.

This means that a fast neutron spectrum reactor leads to fewer high mass isotopes compared to a thermal reactor, since the transuranics (TRU) isotopes are more likely to fission

As for point 2, the discussion on the neutron surplus indicates that, again, fast neutron spectra systems should be preferred


For the purpose of transmutation, both critical and sub-critical fast neutron systems can be used.

  • In fact, from a physics point of view, both will offer the same basic features, as required.
  • However, a specific advantage of (any) sub-critical system for transmutation is that these reactors can be loaded with practically any type of fuel and, in particular, a fuel made only with TRU (no U) and a high percentage of the so-called Minor Actinides (MA): Am, Np, Cm etc.
  • In fact most higher mass TRU that should be transmuted, have very low delayed neutron fractions compared to U-238, U-235 or Pu-239:
  • Sub-criticality allows to relax the requirement of a significant delayed neutron fraction, as required for safety purpose.

The transmutation performances of three fast neutron systems, i.e. a critical fast reactor, a Fusion-Fission Hybrid and an ADS will be shortly compared (for a specific fuel cycle scenario) at the end of the next Tutorial.

  • However, advantages and disadvantages of each type of systems depend not only from specific strategies but also from technological feasibility and technological readiness issues.
  • Here, as an introduction to the comparison, we will only shortly present some physics features of a specific « hybrid » system: the ADS (Accelerator Driven System). More both on Fusion-Fission Hybrids and ADS in other presentations!










(1-f) : to the grid








multiplying system






External Source

For a sub-critical system (Keff < 1), the condition to have a stationarysystem is heuristically written as follows:



prompt neutrons/fission)


For an ADS:







is the number of neutrons which come back to the subcritical core if all the fission energy Ef is transformed into proton current:

f : fraction of Ef

used to feed the accelerator.


To evaluate Γ:

If Ep~ 0.5 - 1.5 GeV and for targets such that:

Z = neutrons/proton ~ 20 - 50

= =  neutrons/fission

(with e~ 0.4, p~ 0.5)

The stationary system condition:





An example of ADS parameters

  • Hypothesis
  • Proton beam energy Ep = 600 MeV.
  • Heavy metal spallation target providing
  •  (number of neutrons/fission in the subcritical core due to the external source, if all the energy produced in it is used to feed the accelerator) = 1.06
  • Thermal power of the subcritical core W = 500 MWt

Energy requirements

The fraction f of energy produced in the subcritical core used for feeding the accelerator depends on the subcriticality level:

If  (average number of prompt fission neutrons per fission) = 2.8, one has:

f = 2.6 % if 1 - Keff = 0.01

f = 5.3 % if 1 - Keff = 0.02

f = 13 % if 1 - Keff = 0.05





Current of the proton beam ip

If Ef (energy released/fission) ~ 200 MeV, and W is the core power:

which correspond, respectively to the following power in the proton beam:

The current ip will be increasingly small, when the system will be closer to criticality and when the power of the sub-critical reactor will be low.