“the physics of subcritical multiplying systems”

The Impact of Nuclear Science on Life Science
The Physics of Sub-critical Multiplying Systems

M. Salvatores

CEA - DEN/DDIN - Cadarache

1. Introduction
The role of accelerator-driven systems has been recognized world-wide [1, 2, 3] as a potential highly relevant tool in order to transmute very large amounts of radioactive wastes and, consequently, to lower the burden on a deep geological storage.
Both the two major variants of the strategies of transmutation which make use of ADS [4], call for sub-critical cores with a fast neutron spectrum and a fuel dominated by mixtures of Pu and Minor Actinides (MA), essentially fertile-free.
These cores are characterised by a very low fraction of delayed neutrons and by a low (even near zero) Doppler reactivity coefficient. In principle, the sub-criticality will help to reduce (or to eliminate) the negative consequences of these characteristics on the safety of the multiplying medium. This is one of the point that will be discussed in the present article.
In fact, the physics of the ADS and of its sub-critical core is well understood, and there are several publications which deal extensively with the subject [see for example refs 5 and 6, among many others]. However, several concepts are new and their understanding requires experimental validation.
In the present article, we will focus on a description of the basic physics phenomena in the sub-critical multiplying core, with reference to the coupling phenomena and their impact on the sub-critical core (SC), when needed.
We will also indicate the areas that need particular care for experimental validation and we will quote some ongoing experimental programs and preliminary results.
Finally, the inspection of some “visual” images of SC as they are presently studied in several laboratories, will be the occasion to point out some relevant design-oriented problems of sub-critical cores and their integration in an ADS.

2. The sub-critical multiplying core in stationary regime
2.1 The flux distribution
In a critical system, the condition of balance of neutron production and neutron disappearing at each point of the phase space (E, r, ) is expressed by the Boltzman equation, which can be expressed in matrix form :
(1)
where A is the “disappearance” and P the “production” operators, and  the flux.
In the same system, made sub-critical, the condition to have a stationary system is to have an external source S(E r, ) such that, e.g, the equation (1) can be written as :
(2)
is the solution of the inhomogeneous equation (2). The distribution in space, energy, angle of is obviously different from that of . Of course, approaches as the level of sub-critically becomes smaller and smaller, approaching the critical configuration.
For an ADS, once defined the material properties, the geometry of the system, the relevant cross-sections and the source intensity (in units of neutrons/sec), the distribution of the inhomogeneous flux is fully determined by equation (2).
Relevant integral parameters, characterising the sub-critical core (SC), such as reaction rates can be easily calculated. This allows to evaluate the power deposited in each point of the system, the damage rate, the breeding ratio etc. This is done exactly as in critical systems, characterized by .
2.2 The reactivity of the sub-critical core
It is formally possible to describe a sub-critical system with the introduction of a parameter k_eff which allows to “restore” the balance equation (2) :
(3)
Since has the same distribution as the “critical” flux, this equation is obviously an approximation of the real case, as described by eq. 2.
In order to improve the definition of sub-criticality and to take into account the change in shape of the flux, it has been proposed a different definition of the sub-criticality, by means of a “K-source” K_S. The procedure is to apply the formal balance condition (3) to the inhomogeneous flux equation (2) :
(4)
Integrating and recalling that one obtains :
(5)
2.3 The * parameter
In a sub-critical system, it is of relevance to the understanding of the behaviour of the source-driven sub-critical core, the evaluation of the relative importance of the source neutrons with respect to the fission neutrons generated in the SC.
One introduces a parameter *, which is the ratio of the average “importance”^(a) of the source neutrons and of the average “importance” of fission neutrons. It can be shown that this parameter * is related to k_eff through the following relation :
^(b) (6)
where is the average number of prompt neutrons per fission, and  the average source neutrons per fission. Relation (6) is given in reference [7], where the experimental determination of * is discussed.
The * parameter plays an important role in the ADS performance parameters assessment. In fact in ref. [8], it is shown that the relation among the proton beam current i_p, the power in the SC and its sub-criticality is given by :
(A) (7)
where W is the power of the SC in watts, _f the energy per fission (MeV) and Z is the number of neutrons per incident proton.
It can be seen from (7) that a value of * higher than 1, can reduce proportionally the proton beam current requirement, for a given sub-criticality level. Measurements of * are made in the CEA facility MASURCA in Cadarache, in the frame of the MUSE program [11], which will be described shortly in a successive paragraph.
3. The kinetics of a sub-critical system
3.1 The asymptotic behaviour
The equations which give the kinetic behaviour of a system driven by an external source, are of the type (ref. 9) :
(8)
where C_i are the precursors of delayed neutrons with decay constant _i. _i is the fraction of the total number of delayed neutrons emitted per fission ( _i = ) due to precursors C_i.
_eff is the neutron generation time and  is the reactivity .
In steady state (i.e. if and ), we have :
(9)
A decrease by a factor h of the reactivity (’ = /h) or an increase by a factor h of the source (S’ = hS), induces an instantaneous increase of the power W’ = hW. For example, if, the system is sub-critical corresponding to - 10, a reactivity insertion of + 5, produces a doubling of the power (see Fig. 1). This of course is totally different from the behaviour of a critical system (which becomes prompt critical).
In more general terms, the kinetic behavior of a critical system is characterized by delayed neutrons and their time constants (about 10 s), while the kinetic behaviour of a SC is determined by the time constants related to the external source, in the sense that an instantaneous variation of source has an effect on the time scale of the prompt neutron lifetime (typically of the order of microseconds).
The evolution of the power with time and the related variation of the temperature is associated to the variation of the reactivity (Doppler reactivity effect, fuel expansion reactivity, reactivity due to the material concentrations in the core, including the coolant etc.). These reactivity effects (feed-back reactivity effects) are essential for the safety of a critical reactor. In a sub-critical core, the feedback reactivity effects are of different relevance according to the level of sub-criticality. In fact for a core deeply sub-critical, the dynamic behaviour is dominated by the external source and its variation in time. Closer to criticality, the feedback effects become more important and the behaviour of the core is approaching that of the corresponding critical core.
In a very simplified way, if the core is sub-critical by -10 , a feedback reactivity equal to  1 , induces a  10 % variation of power and a  50 % variation of power if the system is sub-critical by
- 2 . In a critical reactor + 1  reactivity insertion makes the reactor prompt critical and - 1  stops the chain reaction. In view of the definition of an “optimal” level of sub-criticality, it is of high relevance to verify the transition of the behavior of the SC from a “source-dominated” to a
“feed-back dominated” regime.
3.2 Reactivity and loss-of-flow accidents
Fast external reactivity insertions give rise to different consequences in critical or sub-critical core. Examples have been given in Refs [10, 11]. In ref. 10 a 0.55 /s reactivity insertion in a PHENIX fast reactor type core, critical or sub-critical at Keff = 0.95, gives rise (at constant external source level) to the following power and average temperature evolutions :

	Critical core	Sub-critical core (k=0.95)
Delay before fuel fusion	2 s	12 s
Inserted reactivity	1.1 	6.6 
Power increase W’/W	2.2	1.5

In ref. 11, a reactivity of 170 /s is injected in a critical core (W_o = 1 GW), or in the same core made sub-critical at -1 , - 2 , - 3 . The results show that prompt criticality is reached in the critical core after 6 ms with a first power peak of 700 GW at 8.5 ms and a second peak of 500 GW at 13.2 ms. In the sub-critical mode, the peaks are respectively of 530 GW at - 1 , 6 GW at - 2  and 2.2 GW at - 3  (t = 16 ms) (see Figure 2).

The increase in power is considerably slower in a sub-critical system, and the total energy deployed is much smaller.

In the case of loss-of-coolant-flow accidents, references 10 and 11 give simple examples, which show that, in the case of no shut-down of the source, the behavior of a -10  sub-critical system is less favourable, since in a critical system the increase of the coolant temperature is slower and lower due to the feed-back effects. Again, the choice of the level of sub-criticality is relevant, if one takes into account the potentially beneficial effects of the intrinsic characteristics of the core.

This of course, has to be verified for each type of core and associated fuel and coolant. It is obvious from these considerations that the accelerator beam intensity must be coupled to the power level of the SC, so that it can immediately been shut down in case of a power excursion.
3.3 Cores with low Doppler effect
In the case of an ADS dedicated to transmutation, the fuel will be dominated by MA which will have a low Doppler effect, due to the absence of U-238.
The dynamic behaviour of the core will be differently affected by this, according to the level of sub-criticality (see Figure 3). At large sub-criticality, the calculations of the effect of reactivity insertion performed with a “standard” Doppler coefficient K_D, or with a “low” Doppler (K’_D = 0.1 K_D), show no difference in the power or reactivity behaviour. Close to criticality on the contrary, the effect can be significant.
3.4 Choice of the sub-critical level
No final criteria have been established up to now in order to define an “optimal” level of sub-criticality. However previous considerations indicate the relevance to find a compromise between the “source-dominated” and the “feed-back dominated” regimes.
More quantitatively in the case that no control rods are foreseen in the SC, the level of sub-criticality should be such that the core stays sub-critical when going from a “hot” state (i.e. normal operation) to a “cold” state (i.e. reactor shut-down). Since thermal feed-backs induce generally (e.g. in standard fast reactors) a positive reactivity effect (K_FB) going from “hot” to “cold” state, one can require that the “cold” core stay sub-critical even in the case of an accidental reactivity insertion (K_AC), due for example to coolant voidage. In that case the required “K_eff” should respect the following relation :
K_eff + K_FB + K_AC < 1 (10)
During operation, the maximum reactivity insertion () can be higher than K_AC. In that case one has the requirement that :
K_eff + < 1 (11)
Moreover, during the reactor operation, the reactivity varies due to the irradiation (burn-up) of the fuel and its isotopic evolution. In general this reactivity variation K_BU is negative, but in some case (e.g. a fuel made essentially by minor actinides, which act as “fertile” materials, since they are transmuted in more “reactive” elements, as it is the case for example of Am-241), K_BU can be positive. In that case, if the core has no control rods and one does not want to modify the external source, e.g. by changing the current intensity, one should have :
K_eff + + K_BU < 1 (12)
Looking for a compromise among the different criteria indicated above, one has also to consider that a very large sub-criticality may not be necessarily the optimal solution. In fact, besides obvious considerations on the “cost” of a strong external source, a largely sub-critical core has a peaked power distribution, dominated by the source distribution and, therefore very far from the flat behavior in space, required to optimise the fuel irradiation (and, consequently, the fuel “transmutation”).
3.5 Reactivity control and monitoring
The control of reactivity and of the power level in a critical reactor is made essentially using control rods. In principle, the control in an ADS can be made with only the external source. As an example, the variation of reactivity with the fuel burn-up can be compensated with an appropriate change of the beam current intensity. It can also be conceived a similar system to control the reactivity change between “hot” and “cold” states. However, major variations of the current would be necessary. For example in a SC without control rods, which has K_eff = 0.99 in the “cold” state, K_eff = 0.98 in “hot” state at the beginning of irradiation cycle and K_eff = 0.95 at the end of irradiation cycle, the source intensity should change by a factor of approximately 5, to account for both the trip towards nominal power and the operation cycle. In this context, it is clear that the use of control rods to insure at least some of the functions of reactivity control, should be carefully examined.
Moreover, if it is true that in a SC, in particular in a “source dominated” mode, to shut down of the source has an instantaneous effect to reduce power, the inverse effect, e.g. an “overshoot” due to a sudden increase of the external source, has the consequence of an instantaneous increase of the power. Although more limited than the potential power increase in a critical reactor, such accidental situation should be examined.
Also, when the reactor is shut down, the consequences of the insertion of the full “reserve” of beam current should be analysed. In fact, if the insertion of the full “reserve” of beam current cannot be excluded, this accidental event could lead to a power variation given by (ref. 12) :
(13)
If W_Max is the maximum allowable power in a short time interval, one can deduce the maximum allowable sub-criticality level such that W’ < W_Max.
Finally, we should mention that in principle long term variations of the reactivity can be achieved by an appropriate variation of the * parameter. This can be obtained, for example, by changing the geometrical arrangement of the buffer (or of the buffer material) surrounding the spallation source.
As for as the monitoring of the sub-criticality level, different methods can be envisaged and experimentally validated. Some examples are as follows :

• 
  Use of a pulse mode of the source. The recording of the time evolution of the counting rates of in-core neutron detectors can allow to measure the reactivity. In fact, the point kinetics predicts the prompt decay of the neutron population after a pulse, to be of the type exp(- t) with
   = ( - )/l_eff. For  and l_eff known one can deduce  from the decay of the neutron population obtained experimentally.
  

• 
  If control rods are foreseen, the modified source multiplication method (MSM, see ref. 13) can be used provided the calibration of the control rods reactivity are performed at near-critical level.
  

3.6 Beam trips
As far as the coupling of the accelerator to the sub-critical core is concerned, one significant point which has been raised [14], is the effect of frequent beam trips on the SC. Since we have seen that the time scale for power variation (due to source variation) is very short, the heat transfer time from fuel to coolant being of the order of 0.1  1 s, the heat is stored in the fuel for ~ 1 s making high thermal conductivity fuels a possible requirement. In a similar way, thermal stresses in the core structures can be expected (due to the difference of time constants between power increase and temperatures variations in the structures) and in the case of frequent beam trips, fatigue failures of the structures could occur and cause safety concerns (see also contribution of M. Napolitano to this issue).
4. Experimental validation
The physics characteristics and the predicted behaviour of a SC, as outlined in previous paragraphs, need an experimental validation, in order to calibrate the calculation tools and to gain confidence in the prediction of the basic safety features of an eventual future ADS, which will be fuelled with very innovative fuels.
The main fields which need experimental validation are :

• 
  the effects of the relative contributions of the source neutrons and of the neutrons generated by fission. * measurements should allow to achieve that objective in stationary conditions.
  

• 
  kinetic experiments performed at different sub-critical levels, with or without feed back effects, can be essential to understand the transition between a “source-dominated” and a “feed-back dominated” regime,
  

• 
  space and energy distributions of neutrons and their variations close to the external source,
  

• 
  sub-criticality level assessment and monitoring,
  

• 
  relation between the external source and the power in the core.
  

4.1 The MUSE Experimental Program - The physical principle
A first experiment to check the physical principles of an ADS, was performed by C. RUBBIA at CERN (FEAT experiment, ref. 15). A proton beam did hit directly a natural Uranium block, and the “energy amplification” was experimentally verified.
Since 1995, at the MASURCA facility of CEA in CADARACHE, a series of experiments called “MUSE” (MUltiplication avec Source Externe) have been performed, (see Table 1) in a collaboration between physicists from Cadarache (CEA) and ISN-Grenoble (IN2P3). The principle of these experiments (ref. 7) is to make the hypothesis of the separability of the effects of the source and of the multiplication in the SC.
Intuitively in fact, one can think that a source neutron, once entered in the sub-critical core, will loose “memory” after 1  2 mean free paths, and will behave as any other neutron produced by fission in the SC.
This hypothesis is of course made for a SC not too largely sub-critical (e.g. with K_eff > 0.95). Under this hypothesis, it is possible to study the neutronics of the source-driven SC, using instead of a true spallation source, a well-known external source. The first MUSE experiments were performed with a Cf-252 spontaneous fission source, located at the centre of a SC (ref. 7).
The present MUSE experiments (MUSE-4) use a pulsed 14 MeV neutron source called GENEPI, built at INS-Grenoble. A deuton accelerator has been coupled to the MASURCA facility, and a deuterium or a tritium target located at the centre of the SC (see Fig. 4). These targets are surrounded by a lead buffer, to simulate the neutron diffusion inside an actual lead (or
lead-bismuth) target. Numerical simulations have shown the validity of the basic hypothesis of the experiments, namely that using a spallation neutron source or the neutrons issued from the (d,t) or (d,t) reactions, the neutron spectrum in the core close to the buffer region is very much the same, whatever the neutron source energy distribution.
This result is shown in Fig. 5, where the neutron spectra are shown at the interface buffer/core and at 10 cm from that interface. Only at the buffer/core interface some differences are observed.
4.2 The MUSE Experimental Results and Techniques
Experimental results of relevance have already been obtained. For example, in Figures 6 and 7, the flux distribution inside the SC of the MUSE-1 configuration is shown in terms of the measured U-235 fission rate, in presence of the Cf-252 source.
Figure 6 shows the radial flux distribution in the core with an without external source. The presence of the source gives a more “peaked” distribution as expected.
Figure 7 shows the axial distributions when the source is at the core/upper reflector interface (+ 25 cm from core midplane). Three axial distributions of the fission rates are shown. Far away from the source, the axial profile becomes less sensitive to the asymmetrical position of the source.
Moreover * measurements have been performed and Table 2 gives the comparison of calculated and experimental results.
Finally, in the configuration MUSE-3, the SC was driven by a (d,t) neutron source GENEPI and the counting rate evolution in time of in-core detectors after a neutron pulse at different sub-critical levels are shown in Figure 8. From the decay rate, the sub-criticality level can be deduced.
In fact, from a short neutron burst (of the order of a few sec), the prompt decay rate in time of the neutron population allows the determination of  (see above the description of pulse mode in § 3.5).
More experiments are planned in the new configuration with the GENEPI accelerator (MUSE-4, see Figure 4), and different experimental techniques (transfer function, MSM method, Rossi- and Feynman-, ref. 16) will be used in order to measure the sub-criticality level, but also and _eff and control rod worths in the SC.

5. Some ADS “images” and technological problems
Several countries and leading research laboratories are actively working on the development of ADS, in particular in the frame of radioactive waste minimisation strategies.
Conceptual designs have been developed. A typical example is the Energy Amplifier proposed by C. RUBBIA [17] and mentioned in the introductory paper by H. Condé.
Conceptual designs have also been developed for experimental ADS, in the power range
80 - 100 MWt. Figure 9 indicates two of these configurations, one, lead-cooled, developed at ANSALDO-Italy, and one, gas-cooled, developed at FRAMATOME/NOVATOME-France.
All these conceptual lay-outs are of a preliminary nature and some relevant technological problems are still to be accounted for in a satisfactory way.
This is the case, for example, of the shielding configurations in the upper part of the systems. The shielding in fact should account for the potential deep penetration of high energy neutrons (E_n > 100 MeV) issued from the spallation of protons (typically E_p = 0.6 - 1.5 GeV).
High energy neutron penetration experimental studies performed in Japan, confirm the very large thickness of material (like concrete or stainless steel) needed in order to reduce to acceptable level the doses around the structures.
The beam entrance configuration is also a matter of concern. In fact, a simple vertical entrance of the beam can imply a very complicated system for the fuel loading-unloading system and can also be not optimal with respect to the need to garantee the beam tube void from the intrusion of back-scattered neutrons.
These are just a few examples of technological problems that can have impact on the coupling of the different components of an ADS, and which could need substantial efforts in order to develop a robust ADS design.
6. Conclusions
The physics of a sub-critical core, driven by an external neutron source, is generally
well-understood.
However, its specific features have never been fully experimentally demonstrated.
Several steps have been undertaken in that direction and planned experiments (like the MUSE experiments), should give most of the demonstrations needed in order to proceed to a sound design of an experimental ADS, as proposed, e.g, in the European Roadmap towards and ADS demonstration [1].
7. References
1. “A European Roadmap for Developing Accelerator Driven System (ADS) for Nuclear Waste Incineration” - ETWG, May 2001.
2. “A Roadmap for Developing Accelerator Transmutation of Wastes (ATW) Technology”

Report to Congress DOE/RW-0519, October 1999.

3. T. MUKAYAMA and al. - “Importance of the Double Strata Fuel Cycle for Minor Actinide Transmutation” - Proc. 3rd OECD NEA Int. Inf. - Exchange Meeting on Partitioning and Transmutation (1994).
4. H. CONDE, this issue.
5. C. RUBBIA and al. - “An Energy Amplifier for Cleaner and Inexaustible Nuclear Energy Production Driven by a Particle Accelerator” - CERN/AT/93-47 (ET) 1993.
6. M. SALVATORES - “Accelerator Driven Systems : Physics Principles and Specificities” -

J. Phys. IV France 9, p. 17-33 (1999).

7. M. SALVATORES and al. - “MUSE-1” : A first Experiment at MASURCA to validate the physics of sub-critical multiplying systems relevant to ADS “ - 2nd ADTT Conference, Kalmar, Sweden, June 1996.
8. M. SALVATORES - I. SLESSAREV - M. UEMATSU - Nucl. Sci, Eng, 116, 1, (1994).
9. G. BELL - S. GLASSTONE - “Nuclear Reactor Theory” - Van Nostrand (1970).
10. M. VANIER - Private Communication.
11. H. RIEF - H. TAKAHASHI - “The transient behaviour of Accelerator Driven Sub-critical Systems” - Int. Meeting - “8ème journées SATURNE” - May 1994.
12. A. GANDINI - M. SALVATORES - I. SLESSAREV - “Coupling of reactor power with accelerator current in ADS systems” - Ann. Nucl. Energy, 27, (2000), 1147.
13. S. CARPENTER - ‘”Measurements of Control Rod Worths using ZPPR” - Proc. Spec. Meeting on Control Rod Measurements Techniques” - Cadarache, April 1976 -

NEACRP-U-75.

14. See Proc. Int. Spec. Meeting - “Utilisation and Reliability of High Power Proton Acceleators”

Mito, Japan, October 1998, OECD-NEA.

15. S. ANDRIAMONJE and al. - Phys. Rev. Letters B 348 (1995), 697-709.
16. G. IMEL - To be published.
17. C. RUBBIA and al. - “Conceptual design of a fast neutron operated Energy Amplifier”

CERN/AT/95-44 (ET), 1995.

Table 1

The MUSE experiments at MASURCA

	Type of source	Range of sub-criticality	Diffusing buffer around the source
MUSE-1 (1995)	Cf-252 spontaneous fission neutron source	- 1.5 %	None
MUSE-2 (1996)	Cf-252 spontaneous fission neutron source	- 3.0  3.5 % K/K	- Sodium - Steel
MUSE-3 (1998)	Pulsed neutron source from (d,t)	- 0.5  - 6. % %	- Sodium - Steel
MUSE-4 (2000-2001)	Pulsed neutron source from (d,d) and (d,t)	- 1  .4 %	lead

TABLE 2 :

* Measurements in MUSE-1 (Ref. 7)

Configuration	Calculated *	Measured *
Cf-252 source at core center	1.25	1.19  4 %
Cf-252 source at core/blanket axial interface	0.91	0.90  4 %

Figure 1


Figure 2 (From Ref. 11)


Figure 3

Insertion of 1/3 reactivity in 1 second

Behaviour of an ADS “PHENIX Type” at three different levels of sub-criticality (Ref. 10)


K = 0.995

K = 0.98

K = 0.95

K = 0.98

K = 0.995

K = 0.95

Figure 4

The MASURCA installation for the MUSE program

Figure 5

Comparison of the neutron spectra obtained with (D,D)

and (D,T) neutron sources (as in MUSE experiments),

with the reference spallation source, in the same configuration







Figure 8

Dynamic Measurements MUSE3-REF, MUSE3-SC1, MUSE3-SC2, MUSE3-SC3


Figure 9

Sketches of ADS, liquid metal cooled (right) and gas cooled (left) (not to scale)

Representative of the European XADS proposals (ANSALDO and FRAMATOME)


potentially the same fuel assembly

(e.g. SNR-300 S/A with MOX fuel)

(^a) ”importance” of a neutron in (E, r, ) : this function which defines the contribution of a neutron born in (E, r, ) to the asymptotic power level, is the solution of the equation adjoin to (1)

(^b) this relation is directly related to the “energy gain”, as defined in ref. 5.

DEN/DDIN-2001-35