Abstract:
An accelerator-driven subcritical breeding reactor is operated with a neutron multiplication coefficient as large as possible in order to require a small input power from the accelerator, reducing its dimension and hence its cost and complexity. The beam-generated spallation neutron yield then becomes comparable to the fraction of delayed neutrons from the fissioned elements. This can be exploited to ensure an accurate on-line determination of the reactivity. Resulting changes can be adjusted with the help of neutron absorbing control rods and/or variations of the proton current. In addition, the temperature variations during operation can be continuously monitored and adjusted in order to avoid that the subcritical systems approaches too closely the (delayed) criticality condition and that the neutron multiplication coefficient remains within acceptable limits.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    The present invention relates to accelerator-driven systems (ADS). 
         [0002]    In the recent years, considerable interest has grown worldwide for accelerator-driven subcritical reactors for instance for Trans Uranic (TRU) burners and for a Thorium based breeder, the Energy Amplifier (EA) as disclosed in WO 95/12203. 
         [0003]    In a subcritical system, the neutron multiplication is less than one and the additional neutrons which are produced by an external proton accelerator can be used in particular to ensure the operation of the breeding reaction chain. Subcriticality appears particularly advantageous in applications where there is a contribution of the effective delayed neutrons much smaller than in an ordinary pressurized water reactor (PWR), a small or adverse Doppler temperature coefficient and possibly also a positive void coefficient depending on the conditions of the coolant. The subcritical operation is in particular helpful in the case of a fast breeder based on U-238 or a thermal or fast breeder based on Th-232, since two neutrons rather than one are hereby necessary to close the process, one to produce the fissile material and the second to fission the daughter element, only slightly less than the neutron multiplicity due to fission. 
         [0004]    As well known, in a subcritical system the neutron multiplication coefficient k is the average probability that a given neutron of the core may continue the chain reaction. The resulting total average number of secondary neutrons produced by the cascade starting from each incoming external accelerator-driven neutron is then given by k+k 2 +k 3 + . . . =k/(1−k)=−1/ρ, where ρ is the so-called excess reactivity. In a subcritical system, we have ρ&lt;0. 
         [0005]    Two components contribute to the neutron multiplication of a subcritical system, namely the instantaneous contribution k p  due to prompt fission neutrons and the contribution k d  due to the delayed neutrons generated by a tiny fraction β eff  of the fission fragments which generate neutrons several seconds after the occurrence of an initial fission, with k=k p +k d . During their time of delay, these neutrons are not present as free neutrons: they are “pre-stored” in a nucleus (such as Kr-87), and during such a short period they are not subject to any appreciable moderation or absorption. The phenomenon of storage of these neutrons increases the effective response time. 
         [0006]    Table 1 shows the 6 decay families of delayed neutron-emitting fission fragments from U-233, U-235 and Pu-239 fissile Actinides. The exponential lifetime for neutron emission and the fractional fission rate are given for each family, for both fast (unmoderated) neutrons (β fast ) and thermal neutrons (β ther ). They are widely different for each of the initial actinide states. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Exponential families of delayed neutrons 
               
               
                 for different fission nuclei. 
               
             
          
           
               
                   
                   
                 Fast 
                   
                 Thermal 
                   
               
               
                   
                   
                 lifetime 
                 Fast 
                 lifetime 
                 Thermal 
               
               
                 Ele- 
                 Group 
                 (1/λ i ), 
                 relative yield 
                 (1/λ i ), 
                 relative yield 
               
               
                 ment 
                 no. 
                 sec 
                 β i /β 
                 sec 
                 β i /β 
               
               
                   
               
             
          
           
               
                 233-U 
                 1 
                 80.00 
                 0.096 
                 79.37 
                 0.086 
               
               
                   
                 2 
                 27.78 
                 0.208 
                 29.67 
                 0.299 
               
               
                   
                 3 
                 7.25 
                 0.242 
                 7.19 
                 0.252 
               
               
                   
                 4 
                 3.14 
                 0.327 
                 3.08 
                 0.278 
               
               
                   
                 5 
                 0.82 
                 0.087 
                 0.88 
                 0.051 
               
               
                   
                 6 
                 0.32 
                 0.041 
                 0.40 
                 0.034 
               
               
                   
                 sum 
                   
                 β fast  = 0.0026 
                   
                 β ther  = 0.0026 
               
               
                 235-U 
                 1 
                 78.74 
                 0.038 
                 80.65 
                 0.0330 
               
               
                   
                 2 
                 31.55 
                 0.213 
                 32.79 
                 0.2190 
               
               
                   
                 3 
                 8.70 
                 0.188 
                 9.01 
                 0.1960 
               
               
                   
                 4 
                 3.22 
                 0.407 
                 3.32 
                 0.3950 
               
               
                   
                 5 
                 0.71 
                 0.128 
                 0.88 
                 0.1150 
               
               
                   
                 6 
                 0.26 
                 0.026 
                 0.33 
                 0.0420 
               
               
                   
                 sum 
                   
                 β fast  = 0.0064 
                   
                 β ther  = 0.0067 
               
               
                 239-Pu 
                 1 
                 77.52 
                 0.038 
                 78.13 
                 0.035 
               
               
                   
                 2 
                 32.15 
                 0.280 
                 33.22 
                 0.298 
               
               
                   
                 3 
                 7.46 
                 0.216 
                 8.06 
                 0.211 
               
               
                   
                 4 
                 3.02 
                 0.328 
                 3.08 
                 0.326 
               
               
                   
                 5 
                 0.79 
                 0.103 
                 0.89 
                 0.086 
               
               
                   
                 6 
                 0.31 
                 0.035 
                 0.37 
                 0.044 
               
               
                   
                 sum 
                   
                 β fast  = 0.0020 
                   
                 β ther  = 0.0022 
               
               
                   
               
             
          
         
       
     
         [0007]    As shown in Table 1 in the case of the fissionable elements U-233, Pu-239 and U-235, the period of the characteristic families of delayed neutrons range from about one second to about one minute depending on the composition of the fissile Actinides. Of course, irrespectively of the source, both beam-driven external neutrons and delayed neutrons undergo subsequent neutron multiplication. 
         [0008]    Various units of reactivity are of common use, amongst them the “dollar” ($). The dollar is defined as the variation Δk of the neutron multiplication coefficient k which is just equal to the contribution of delayed neutrons, namely Δk=β eff , with the “cent” being the hundredth part of a dollar. Actual values are 1 $=2.1×10 −3  for U-233, 1 $=2.9×10 −3  for Pu-239 and 1 $=7.3×10 −3  for U-235. 
         [0009]    As can be seen, each isotope contributes a differing number of delayed neutrons to any practical fuel mixture. It is therefore helpful to define a value of β eff  for the reactor system: 
         [0000]      β eff =Σ i=1   n   P   eff ( i )×β eff ( i )
   where: P eff (i) is the fractional contribution to total fission of isotope (i) in a mixture of n isotopes present in the reactor. It is noted that P eff (i) varies with time due to reactor operation and radioactive decay;
       β eff (i) is the contribution of delayed neutrons for isotope (i).   
       
 
         [0012]    The theory of subcritical systems for nuclear fuels has been extensively discussed since the fifties. More recently, with the advent of the Energy Amplifier and ADS concepts, a number of nearly zero power subcritical arrangements have been extensively studied experimentally, amongst which the experiments FEAT at CERN (Switzerland), MASURCA in Cadarache (France) and YALINA in Minsk (Belarus). 
         [0013]    In these experiments, a number of methods to identify the reactivity-related parameters have been described in detail. These methods have been generally based on the use of a narrowly pulsed, low-power beam source (for instance of Dirac-like shape). They are involving extremely small amounts of fission power and no appreciable temperature variations. They are adequate for calibration purposes and reactivity estimates. But these studies are not entirely adequate to define a practical reactivity monitoring and the necessary feedback control procedures for the operation of a commercial, high-power accelerator-driven system where instead the beam power is large and continuous, i.e. the analogue of the regulation methods of an ordinary critical reactor. 
         [0014]    Unlike the above-mentioned nearly zero-power subcritical arrangements, in commercially oriented larger power generating plants (either critical or subcritical), temperature variations play a fundamental role. In a high-power system the most relevant reactivity feedback mechanism is the Doppler effect, which depends on the instantaneous temperature distribution of the fuel of the core. When materials heat up, resonances in the reaction cross sections get wider, thus changing the probability for the reactions to occur and therefore altering the neutron multiplication coefficient k. The relevant parameter is the so-called temperature coefficient, defined as Δk T =dk/dT (in units of K −1 ), where T is the absolute temperature of each fuel element. The actual value of          Δk T           =         dk/dT         , suitably averaged over the reactor volume, is strongly dependent on the nature of the elements that constitute of the core. Its value may be either positive or negative depending on the composition and the geometry of the fuel and of the coolant materials. 
         [0015]    Temperature variations are also affecting the behaviour of the coolant and of the whole geometrical structure of the core. When the temperature increases, there is decrease of the coolant density ρ(T), with correspondingly fewer neutrons captured and a geometrical expansion in the lattice geometry. The neutron spectrum and therefore the neutron multiplication coefficient k is correspondingly modified by the so-called void coefficient dk/(dρ/ρ), which introduces changes summed to the temperature coefficient dk/dT, and by changes of the “buckling” due to geometrical expansion. 
         [0016]    In an ordinary critical reactor, the timely change (not its actual value) of the produced power is controllable by adjusting the neutron multiplication coefficient k as a function of time. The reactor is then operated in a self-generating power critical mode with k≈1. The operator determines directly the direction and the rate of motion of the control rods, that is, effectively, the second derivative of the power level. The instantaneous k excess above 1 must never exceed 1 $ since a critical reactor is controllable only within the extent of the contribution of the delayed neutrons, where the rate of power change remains acceptably slow. In these conditions, the rate of change is determined by the presence of the delayed neutrons in order to allow enough time to adjust mechanically with control rods the self produced power. For instance, in a standard PWR, a criticality constant of 1/1000 above k=1 would increase the neutron population by about 0.9 percent per second, leaving ample time to correct the criticality factor with the help of control bars before an undue increase in the reaction rate takes place. If (k−1) exceeds the entire contribution due to delayed neutrons, the reactor becomes “prompt” critical, with the most dramatic consequences. The lifetime of a neutron will be typically of the order of a few μsec, and for a criticality factor exceeding the prompt value by as little as 1/1000, the multiplication rate will increase by a factor (1.001) 1000 ≈2×10 4  at each second! 
         [0017]    In a subcritical reactor, the fission power is driven directly by the proton beam current. This corresponds to a precise proportionality between the thermal fission produced power P therm  and the external accelerator beam power P beam =i beam ×T p /e, i beam  being the proton current in Ampere, e being the proton elementary charge and T p  being the kinetic energy in eV. The multiplying factor between P therm  and P beam  is k/(1−k) as generated by the nuclear cascade. 
         [0018]    The development of modern accelerators has permitted the production of a substantial neutron flux with the help of a proton-driven high-energy spallation source. The spallation in a heavy Z target, like for instance molten Lead, may produce as many as 30 neutrons/proton for a beam made of protons having a kinetic energy of 1 GeV. Accelerators in this energy domain, for instance either cyclotrons or superconducting linear accelerators (LINAC), may produce a beam power P beam  that is as much as 50% of its required primary electricity supply, thus requiring from the accelerator only a modest fraction of the electricity which is generated by the reactor. 
         [0019]    In previous subcritical projects, the value of the criticality k has been chosen far enough from unity, typically k≈0.97 or even smaller, but sufficient to ensure that an adequate subcriticality margin is guaranteed a priori even under the most exceptional adverse conditions. A “scram” mechanism is needed only in the case of an emergency shutdown and when the reactor is kept off for a very extended period of time. 
         [0020]    In these conditions (k 0.97), a subcritical EA traditionally operated at a thermal power of 1.5 GWatt requires for instance a current i beam  of about 24 mA for a proton beam of 1 GeV kinetic energy on a Lead spallation target. This represents a substantial technology advance both for the accelerator itself and for the spallation target. 
         [0021]    It is an object of this invention to propose an alternative mode of operation which makes it possible to effectively control the reactor in a high power mode (suitable for commercial energy production plants), without the requirement for a very large accelerator. 
         [0022]    It is proposed here to operate in subcritical conditions with a value of the neutron multiplication coefficient k as large as possible in order to require a smaller power from the accelerator, reducing its dimensions and hence its cost and complexity. 
         [0023]    A method of operating an ADS in subcritical conditions is disclosed. 
         [0024]    The method comprises:
       directing accelerated particles onto a spallation target;   multiplying neutrons from the spallation target in a core loaded with nuclear fuel comprising fissile and fertile material; and   controlling reactivity in the core such that an effective neutron multiplication coefficient is maintained in a range above 0.98.       
 
         [0028]    It has been found that operating at such large values of the effective neutron multiplication coefficient k eff  (typically one or a few $ below criticality) introduces a completely different phenomenology. Unlike in the traditionally recommended case of k≈0.97, the amount of beam-generated spallation neutrons becomes comparable to the fraction β eff  due the number of delayed neutrons. Hence, they both contribute in a comparable way to the multiplying cascade. Any change of the proton current will induce a sizeable modulation in the resulting fission power, with the characteristic time pattern of the delayed neutrons, and allow an accurate on-line estimation of the neutron multiplication coefficient k or reactivity ρ. 
         [0029]    The effective neutron multiplication coefficient k eff  can thus be controlled to be compatible with (a) the maximum value of k with which the subcritical reactor may be brought to operate and (b) the largest value of k which can be safely maintained. 
         [0030]    For instance, a fast U-233 fission-driven subcritical reactor with (1−k)=3.15×10 −3  and a thermal power P therm  of 1.5 GWatt can be controlled by a proton beam of 1 GeV kinetic energy with a current i beam =2.0 mA interacting in a molten Lead target, well within the present status of the art of the accelerator technology. The overall power gain of the accelerator at (1−k)=3.15×10 −3  is then typically G=P therm /P beam =750. 
         [0031]    In general, the present method is well suited to commercial energy production applications with a reasonable accelerator beam power, for example in a range of 0.5 to 5 MWatt 
         [0032]    In an embodiment, the above-mentioned range for the effective neutron multiplication coefficient is above 0.99 and below 0.999, thus providing very high power gains in the system. 
         [0033]    The reactivity in the core is advantageously controlled in a range above−4$, where the reactivity unit ‘$’ is for the reactor system. A preferred control range for the reactivity in dollars between −3$ and −0.5$. 
         [0034]    The operation in the very high k-range provides suitable conditions for all the main measurements necessary for the regulating control of the accelerator-driven subcritical core of a commercially oriented larger power generating plant. As in most existing nuclear reactors, neutron counters are typically distributed in the core. Controlling reactivity in the core may then include:
       applying a step change to reduce a beam current of the accelerated particles;   measuring a variation of a neutron counting rate provided by the neutron counters in response to the step change of the beam current;   estimating a drop of the counting rate related to the loss of prompt neutrons due to said step change; and   evaluating a ratio of the estimated drop of the counting rate to a value of the counting rate before said step change.       
 
         [0039]    In order to a obtain a statistically reliable estimation of the drop of the neutron counting rate even in the presence of temperature variations, the variation of the counting rate is advantageously extrapolated from after the step change of the beam current towards the time of said step change. The extrapolated value at the time of the step change gives an indication of the relative level of the delayed neutrons in the steady regime immediately preceding the step change, which is directly related to the reactivity p in dollars. It is noted that the amount of neutrons just after the step change of the beam current can generally not be measured directly because the counters require sufficient counting statistics which cannot be practically achieved because such “semi-stable” level is not maintained long enough due (i) to the decay of the delayed neutrons associated with the lost spallation neutrons which do no cause new fissions any more after the step change and (ii) to the changes in the multiplication coefficient induced by the temperature change inside the core. 
         [0040]    In particular, a period following the step change, in which the beam current is kept at the reduced value and the variation of the neutron counting rate is measured for extrapolation, may be more than 100 milliseconds or, preferably, more than 1 second. In order to limit the thermal stress in the core structure, the step change corresponds to a fraction only of the beam current. Typically, the step change reduces the beam current by less than 50%. 
         [0041]    In a practical implementation of the method, the accelerated particles directed onto the spallation target are in the form of a continuous particle beam. A pulsed beam is not well suited because a too frequently repeated sudden (in μs) switching off of the full proton beam even for a relatively short period of time is hardly applicable to any large reactor due to the safety requirements related to an excessive number of repeated thermal shocks of the core structure. 
         [0042]    In an accelerator-driven subcritical system, the main controlling element is the variability of the proton current. The variations of the current should have generally an acceptably small rate of change and should be whenever possible of limited amplitude. So the particle beam is preferably operated at a nominal beam current, except in phases of estimating reactivity in the core, and the reactivity control comprises adjusting the position of neutron-absorbing control elements in the core. 
         [0043]    The largest value of k which can be maintained is related to the growth in k due to beam variations and it is generally associated with the (negative) temperature coefficients of the core. In particular, the operating value of k should be sufficiently far away from criticality in order ensure that the inevitable sudden loss of the beam due to an accidental failure of the accelerator, the so-called “trip”, never exceeds the condition k=1. It is desirable to detect beam “trips” very quickly so that corrective emergency measures can be taken to prevent criticality. The temperature drop of the fuel pins after the occurrence of an exceptional sudden “trip” due to a proton beam loss is relatively fast, with a primary decay time constant of only a few seconds. The sudden occurrence of an unexpected change or loss of the proton current will automatically activate the prompt insertion of fast moving control rods with a reactivity reduction (“scram”) such as to bring the reactor well away from potential criticality conditions even before the main change in the temperature of the fuel rods of which the reactor is made has occurred. Thus, an embodiment of the method comprises: detecting any interruption of the accelerated particles; and in response to detection of an interruption, inserting scram neutron absorbers into the core. Preferably, the scram neutron absorbers are inserted into the core after a period of more than 100 milliseconds, preferably more than 1 second, following detection of an interruption of the accelerated particles. A variation of a neutron counting rate provided by neutron counters distributed in the core can be measured in such period. A drop of the counting rate related to the loss of prompt neutrons due to the interruption is then estimated to derive a reactivity value based on a ratio of the estimated drop of the counting rate to a value of the counting rate before the interruption. 
         [0044]    In order to maintain stable operating conditions of the core while operating the particle beam at a nominal beam current (except in phases of estimating reactivity in the core), the reactivity control may comprise:
       continuously monitoring a neutron counting rate provided by neutron counters distributed in the core; and   in response to detection of a deviation condition of the monitored counting rate, performing a phase of estimating reactivity in the core.       
 
         [0047]    This makes it possible to check stability of the operation and to obtain a reactivity estimate (with the related beam reductions causing thermal variations) only when necessary to make sure whether some action to correct the reactivity (such as moving control rods) is needed or not. 
         [0048]    Another option, which can be combined with the previous one, is to carry out a reactivity estimation phase periodically. However, the periodicity of such phases should be sufficient (typically more than an hour) to minimize thermal stress to the core. 
         [0049]    In addition, the accidental occurrence of a beam “trip” should preferably be very limited (as compared to the ordinary conditions in a conventional proton accelerator), in order to avoid repeated thermal shocks of the structure of the core. This can be prevented by introducing appropriate additions in all the components related to the accelerator. For example, the accelerated particles can be provided by an accelerator complex having redundant components to ensure continuity of the beam current. 
         [0050]    In particular, the accelerator complex may have at least one accelerating structure with a plurality of serially-mounted accelerating cavities to apply respective energy gains. If the energy gain of one of the cavities is lost, the lost gain is redistributed between the other cavities using accelerating RF-phase angles. The accelerator complex may also have a plurality of accelerating structures, and in case of failure of one of the accelerating structures, the beam current of at least one other accelerating structure is increased to maintain the overall current of accelerated particles. In an embodiment, two particle beams from the plurality of accelerating structures are merged side-to-side upstream of the spallation target using a magnetic structure and a septum. Those two particle beams may include a first proton beam and a negative ion beam having the same kinetic energy, electrons being stripped from the negative ion beam to provide a second proton beam merged with the first proton beam upstream of the spallation target. 
         [0051]    Another aspect of the invention relates to a subcritical accelerator-driven nuclear system, comprising:
       at least one particle accelerator (one in use with possible additional standby redundancy as required);   a spallation target receiving the accelerated particles;   a core adjacent to the spallation target, loaded with nuclear fuel comprising fertile material;   a coolant circuit for recovering heat from the core;   neutron counters distributed in the core; and   a control system cooperating with the neutron counters for controlling reactivity such that an effective neutron multiplication coefficient is maintained in a range above 0.98.       
 
         [0058]    Other features and advantages of the method and system disclosed herein will become apparent from the following description of non-limiting embodiments, with reference to the appended drawings. 
     
    
     
       BRIEF DESCRIPTION THE DRAWINGS 
         [0059]      FIG. 1  is a schematic diagram of a subcritical reactor core which can be used to carry out a method in accordance with the invention. 
           [0060]      FIG. 2  is a diagram representing a typical spallation neutron yield for each incident proton on a thick molten lead target as a function of the proton energy. 
           [0061]      FIG. 3  shows diagrammatically an example of proton accelerator complex including duplicate standby unit which can be used to implement the invention. 
           [0062]      FIG. 4  illustrates an alternative example of the accelerator complex. 
           [0063]      FIG. 5  represents graphically the contribution of surviving delayed neutron rate immediately after a step reduction of the proton current as a function of the number of $ away from delayed criticality. 
           [0064]      FIG. 6  is a diagram representing different operational conditions as a function of the total neutron multiplication coefficient k, or the number of $, in the case of a breeder reactor based on U-233. 
           [0065]      FIG. 7  represents graphically the neutron counting rate in arbitrary units (a.u.) as a function of time after a beam “trip” at time t=0. 
           [0066]      FIG. 8  represents graphically the neutron counting rate in arbitrary units as a function of time after a 30% step change of the beam current at time t=0. 
       
    
    
     DESCRIPTION OF PREFERRED EMBODIMENTS 
       [0067]    The objects, features and advantages of the invention will now be illustrated in more detail with the aid of the following description of the preferred embodiments. Still further objects and advantages will become apparent from the consideration of the ensuing description and accompanying drawings. All those specific examples are intended for purposes of illustration only and are not to limit the scope of the invention. 
         [0068]    In an ADS as illustrated schematically in  FIG. 1 , spallation neutrons are generated in a target  101  located in a central region of a reactor core  100  by directing high energy particles, such as protons having a kinetic energy of the order of 1 GeV, onto heavy nuclei forming the target. Among different materials suitable for spallation targets, Lead is advantageously used because of its high neutron yield when hit by high energy protons. Also Lead in the liquid phase can be used as a coolant to recover thermal power from the core. Other elements including Bismuth have attractive properties to be used as spallation targets. 
         [0069]    The exemplary core  100  shown in  FIG. 1  has an enclosure  102  containing liquid Lead. The central region  101  of the core forming the above-mentioned spallation target is surrounded by fuel assemblies  103 . The nuclear fuel contains fertile elements such as Th-232 or U-238 which can breed fissile elements (U-233 or Pu-239) after capturing neutrons. The fissile element can be fissioned by reacting with another neutron. The prompt and delayed neutrons resulting from the fission reaction, along with new spallation neutrons from the target, continue the breeding and fission process. The overall neutron multiplication coefficient k is kept below 1 to avoid criticality. 
         [0070]    In the configuration illustrated in  FIG. 1 , the fuel assemblies  103  are immersed in molten Lead which is heated by the transfer of kinetic energy from the fission fragments. One or more heat exchangers  104  are provided in the enclosure to recover heat from the Lead coolant. The secondary circuit is for example based on steam to operate a turbine. The incident proton beam  105  enters the central target region  101  of the core through a beam window  106  located at the end of a beam channel  107 . The layout of the core  100  can generally be as described in WO 95/12203 which also explains the relevant physics. 
         [0071]    As in conventional critical reactors, neutrons counters  110  are distributed in the fuel region of the core to continuously obtain neutron count rates indicative of the neutron flux within the core. A control rod system  111  is also provided in the core region in order to adjust the reactivity as described below. Finally, other neutron absorbing rods form a scram absorber system  112  activated to stop the reaction when certain operational conditions are detected. 
         [0072]    A control system (not shown) gathers information from various sensors provided in the accelerator complex and the reactor core, including the neutron counters  110 , to operate the installation, including the accelerator complex, the control rod system  111  and the a scram absorber system  112 . How such control is performed is described further below. 
         [0073]      FIG. 2  shows a curve representing the average number of spallation neutrons produced by a single incoming proton in the illustrative case of a thick target made of molten Lead, as a function of the proton energy. 
         [0074]    A number of state of the art choices are available for the accelerator. The continuous proton intensity can be varied promptly and within wide limits, down to zero if desired, with the help of a control grid in the proton source. 
         [0075]    For indicative purposes, the case of a 1 GeV superconducting LINAC with duplicate redundant unit is described schematically in  FIG. 3 . It will be appreciated that alternative accelerating methods of an equivalent performance can be chosen. The accelerator system shown in  FIG. 3  is of a well-established design. It may be divided into three main segments:
       an injector 1, made out of a source providing protons in an energy range around 10 keV, a radiofrequency quadrupole (RFQ) accelerating the protons up to about 5 MeV, followed by a Drift Tube Linac (DTL), up to a proton energy of indicatively 15 MeV;   an intermediate section  2 , with a DTL structure, either normal or superconducting, to accelerate protons until about 85 MeV;   and finally a superconducting LINAC structure  3  which completes the accelerating process up to the prescribed energy (1 GeV in our illustrative example).       
 
         [0079]    A preferred feature of the otherwise conventional accelerating structure is the requirement of a very small rate of accidental “trips” due to beam failures. Two methods are presented below, respectively based on an appropriate redundancy of the active components and an appropriate duplication of the accelerating structures. 
         [0080]    Redundancy can be realized for every active component of the accelerator. Each accelerating cavity has a RF synchronous phase angle φ s  around which during acceleration individual particles perform longitudinal phase space oscillations. The accidental loss of the RF in one (or maybe more) cavities will maintain the accelerated beam current provided there is sufficient spare RF voltage in order to let the other cavities redistribute spontaneously their required increments of the voltage gain with a correspondingly larger sin(φ s ). 
         [0081]    Duplication consists in the doubling of the complete accelerating structures from the source to the final energy, with two (or maybe more) and totally independent channels, housed in two nearby but separately shielded enclosures. This permits, if needed, the controlled (repair) access to one of the structures when the other one is operating, as shown in  FIG. 3 . Each independent accelerating channel is capable of providing the total required current i beam , although each of them may be normally controlled to operate for instance at i beam /2. The two accelerated proton currents are accurately and continuously measured with independent current transformers, 4 and 5. In the event of an accidental failure (“trip”) of one of the structures, the full current i beam  is taken over in a negligibly short time (of the order of μs) by the other already operating structure. According to well known practice, at the end of the accelerators the two beam transports are merged together side to side for instance with an appropriate magnetic septum  6  and transferred with the help of the common bending and focusing magnetic transport structure  7  to the spallation target  101  inside the subcritical reactor core  100 . The sum of the beam currents is measured at all times by a dedicated, redundant current transformer  10 . 
         [0082]    In an alternative scenario, one of the accelerators is operating with negative ions H − ≡H 0 e − ≡(pe − )e − , and the other one still with protons H + . The two beams with opposite signs are brought magnetically together and a very thin stripping foil is removing the electrons, namely e − , thus producing a uniquely merged proton beam. As shown in  FIG. 4  while the proton beam is measured by the current transformer  4 , another current transformer  11  measures the negative ion current. The two beams are brought together with the help of two separate bending magnets  12 ,  13  and a common magnet  14 . The negative beam is stripped with a thin foil  15  and the resulting proton beam is transported to the spallation target with the help of the (redundant) sum current transformer  10 . 
         [0083]    Similar considerations based on redundancy and duplication apply to any other alternative accelerating method, like for instance the alternative of the cyclotron. 
         [0084]    According to the present invention, three main components inside the reactor provide for the processes necessary to control and adjust the accelerator-driven subcritical core, operated by the extracted proton beam current. They are:
       the scram absorber system  112  to perform a prompt EA shutdown quickly in case of failure of the accelerator current and in particular in the case of an accidental “trip” of the proton beam. This is actuated promptly by inserting fast neutron-absorbing “scram rods” into the core in order to bring down the value of neutron multiplication coefficient k to a safe value. This shutdown should be performed early enough (i.e. of the order of one second) in order to minimize the consequences of the temperature variations especially in the fuel pins or other equivalent structure of the core;   the uniformly distributed array of neutron sensitive counters  110 . Following well-known practice, this kind of counters are only sensitive to neutrons and do not record appreciably other signals, like for instance α, β, γ radiation or other ionizing particles. The N counters of the array are arranged uniformly inside the core in order to record the neutron counting rates dC i /dt, i=1, . . . , N. With the fission process being the dominant power-generating process, the appropriately weighted sum of the combined neutron counting rates       
 
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                  
                 t 
               
             
           
         
       
     
         [0000]    is directly proportional to the instantaneously produced thermal power of the core. Therefore the rather indirect measurement of the actual instantaneous power can be substituted at all times by a measurement of the in situ counter array. A high level of redundancy is recommended in the combined neutron counting rates: this is normally performed with agreement for instance between two out of three duplicated channel arrays.
       the control rod system  111  which provides an appropriate number of neutron-absorbing devices distributed over the volume of the reactor core (control rods) in order to introduce, with the help of fine mechanical movements, the required changes of the neutron multiplication coefficient k.       
 
         [0088]    The last two items closely resemble the ones of an ordinary critical reactor, although their applicability is quite different since here they are intended for the operation of a subcritical reactor, aided by the nuclear fission energy coming from the external neutron source supplied by a suitable particle accelerator. 
         [0089]    Several different and complementary procedures can be performed with the help of the above-mentioned systems. Combining these procedures provides for measurements useful for the operation and control of the accelerator-driven subcritical core. 
         [0090]    A first and continuously running procedure relates to the stable operation of the subcritical reactor. Extensive experience with critical reactors, which is readily extended to the subcritical operation driven by an external spallation source, has shown that reactors may normally run in steady conditions at a constant power for several hours without the necessity of changes in the position of the control elements. Causes and effects of deviations from the steady state behavior can be either momentary or extended because of some change in the system temperature, proton current, coolant flow or load and so on. They may develop slowly over a long period of time because for instance of the fuel burn-up and accumulation of fission products in it. If the reactor power is to be held constant, some means of compensating for changes of the k value are necessary. Compensation for these changes is often self-regulated by the reactor itself. 
         [0091]    In these normal conditions, the proton accelerator current is kept at its nominal value and the neutron counting rate dC/dt is continuously recorded as a function of time, in order to alert for its possible variations. It is generally expected that the combined neutron counting rate (and hence the thermal fission produced power P therm ) will remain very close to a the pre-assigned value, without significant changes in the position of the control elements, which may be however slightly adjusted whenever necessary with the help of the small mechanical movements of the neutron-absorbing control rods. In particular, the contributions to k coming from the temperature variations in the core  100  should remain nearly constant as long as the system temperature, coolant flow or load remain sufficiently stable in order to be automatically regulated by the control rods of the reactor. 
         [0092]    Whenever a significant change of the neutron counting rate occurs, or periodically, a phase of estimating reactivity in the core is performed following an adequate procedure described below, with the main aim of restoring the prescribed conditions and ensuring that the neutron multiplication coefficient is safely away from criticality under any circumstance. 
         [0093]    It is necessary to activate controlled changes of the proton current, for instance in order to turn on or off the reactor power or to adjust it to the level required for electricity generation. A rare but inevitable event is the total loss of the proton current. Switching on or off the full proton beam systematically even for a very short time (even milliseconds) is to be considered an exceptional event which however must be very carefully considered. 
         [0094]    Even for a few seconds, any change in the proton current will imply corresponding changes in the temperature of the fuel of the core and therefore changes in the average temperature coefficient          Δk T           =         dk/dT         , suitably averaged over the reactor volume, in the void coefficient of the coolant dk/(dρ/ρ) and in the expansion of the structure of the core. The different characteristic time constants of these phenomena due to thermal changes must be experimentally identified and separated out from the effects due to the delayed multiplication coefficient k d . 
         [0095]    In order to describe a variation in the proton current, we decompose the effect into a component of the proton current that is remaining constant and a (smaller) amplitude which is changing as a step function. 
         [0096]    A sudden switching off of the entire beam current would in fact cause a major temperature variation of most if not all the components of the reactor, especially of the fuel material inside the rods. Thus, it should be discouraged as a routine action. On the other hand, in view of the high rate and the consequent high statistical precision of the neutron counters, even a relatively small change of the counting rate can be precisely evaluated. 
         [0097]    After a prompt stepwise change of the proton beam current, we can identify, in the neutron counting rate as measured, contributions of the neutron multiplication coefficient k due to (A) the prompt fission neutrons k p , (B) the delayed neutrons k d , generated by the fission fragments and (C) the variations due to the effects of the temperature k temp . Each one of the three effects has its own specific time dependence which is discussed below. 
         [0098]    (A) The fast component of the nuclear cascade will be quickly switched off by the indicated step function of the proton current. According to the point reactor kinetics model, valid to a first approximation for k d  near 1, the decay of the neutron population is characterized by a fast exponential decay with a time constant α=(1−k P )/Λ where k P  is the prompt neutron multiplication coefficient and Λ≈1 μs is the mean prompt lifetime. Hence the measurement of a can be used to infer k g  provided Λ is known and α is constant. In reality, the value of α strongly deviates from being constant since it reflects the presence of the time-ordered neutron lethargy as a function of the neutron energy and the complicated cross-sections as a function of the neutron energy. Evaluating this very fast change has been already proposed to determine the prompt multiplication coefficient from the experimental observation of the time variation of the parameter α by the so-called k g -method (see A. Billebaud et al. “ Prompt multiplication factor measurements in subcritical systems: From MUSE experiment to a demonstration ADS ”, Progress in Nuclear Energy, 49 (2007), pp. 142-160). It requires that Λ is known a priori from a variety of different k p  values, for instance with the help of Monte Carlo calculation provided the actual fuel composition is introduced. In addition, since the transition is very fast, occurring within less than 1 ms, a huge counting rate dC/dt is necessary in order to determine with sufficient statistical accuracy the decay distribution in this short time. This method is not considered as immediately applicable to our case. 
         [0099]    (B) The effects of delayed neutrons, generated by the fission fragments are considered next. To this effect, the observation of the counting rate R=dC/dt is continued for some time, typically a few seconds, until a semi-stable level is reached characterized by the survival and subsequent decay of the delayed neutrons. Let R 0 +RB be the rate prior to the step change of the proton beam, where R 0  is the contribution to the neutron counting rate associated with the fraction of the beam which is cancelled by the step function and RB the rate due to the unchanged beam component. Let R 1  be the surviving contribution of R 0  due to the semi-stable level of the delayed neutrons. Note that the delayed neutrons (like the spallation neutrons) are also multiplied by the neutron multiplication coefficient k. The resulting reactivity ρ/β eff  in units of $, where ρ=(k−1)/k, can be evaluated using: 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                      
                     
                         
                     
                      
                     1 
                   
                   = 
                   
                     
                       R 
                        
                       
                           
                       
                        
                       0 
                     
                     
                       1 
                       - 
                       
                         ρ 
                         / 
                         
                           β 
                           eff 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0100]      FIG. 5  represents the contribution of the surviving delayed neutron rate immediately after a step reduction of the proton current Δi/i equal to 1, 0.5, 0.3 and 0.15, as a function of the number of $ away from delayed criticality. When approaching smaller values of $, the effect of the surviving delayed neutron rate is progressively increased. 
         [0101]    In a case using U-233 as the fissile isotope,  FIG. 5  shows in the abscissa the contribution R 1 /(R 0 +RB) due to the semi-stable (initial) level of the delayed neutrons, and in the ordinate the k-value both in size and in $ from (delayed) criticality. Four curves  20 ,  21 ,  22  and  23  are shown, corresponding to Δi beam /i beam =R 0 /(R 0 +RB)=1, 0.5, 0.3 and 0.15, respectively, namely decreasing values of the step in the proton beam, where Δi beam  is the magnitude of the step change of the beam current, and i beam  the value of the beam current just before the step change. As previously, R 0  is the fraction of the initial proton beam which undergoes the step function to zero, and RB the fraction due to the unchanged beam component. The k-value corresponding to 1.5 $ from (delayed) criticality is shown by the dashed line  24 . 
         [0102]    As shown in  FIG. 5 , with k approaching 1, the relative contribution due to the delayed neutrons is growing in size. For instance, for (1−k)=1.5 $, the fractional delayed neutron semi-stable plateau is R 1 /(R 0 +RB)=0.12 for Δi beam /i beam =0.3, increasing for Δi beam /i beam =0.5 to R 1 /(R 0 +RB)=0.20 and decreasing for Δi beam /i beam =0.15 to R 1 /(R 0 +RB)=0.06. The signal R 1 /(R 0 +RB)=(0.120±0.005) will give an uncertainty in (1−k)=(1.5±0.1) $. 
         [0103]    It appears from  FIG. 5  that the sensitivity to the effect due to the semi-stable (initial) level of the delayed neutrons is much less significant for smaller k values. For instance, in a traditional subcritical system with k=0.975 and again Δi beam /i beam =0.3, the delayed neutron signal will be much smaller, i.e. R 1 /(R 0 +RB)=(0.0259±0.005), leading to a much higher uncertainty on the neutron multiplication coefficient with a rather large measured uncertainty in the energy gain, G=96 −17   +26 . 
         [0104]    From  FIG. 5 , it can be determined that the value of the effective neutron multiplication factor should be in a range above 0.98 (and below 1 to remain subcritical of course), and preferably in a range above 0.99 and below 0.999. An operational diagram as shown in  FIG. 6  can be derived. Any value k≧1 (or ρ&gt;0) must be avoided to prevent criticality, with k&gt;0.98 or 0.99 to ensure sufficient sensitivity to monitor the reactivity p. In the case of  FIG. 6 , we have set a subcriticality value of −1.5 $ for operation of the reactor, corresponding to line  24  in  FIG. 5  (k≈0.9965 in the case of U-233). As long as p remains below −0.5 $ (k≦≈0.999), the operating conditions are not abnormal. 
         [0105]    The operational range of the reactor may also be defined in terms of dollars, i.e. reactivity values (like conventional critical reactors). This is convenient since the dollar values are actually monitored and the translation to k-values depends on the specific kind of fissile isotope(s) being used in the core. Based on  FIGS. 5 and 6 , the range for ρ is advantageously above −4.0 $, and a typical range will be between −3.0 and −0.5 $. 
         [0106]    (C) Finally, the effects due to the temperature variations are discussed. As already pointed, out any (sudden) variation in the proton current will cause variations of the fuel temperature and consequently a variation of the neutron multiplication coefficient k. These variations are dependent on the actual structure of the subcritical reactor and they may vary substantially according to the situation. Most of the scenarios considered so far are characterized by a small and negative overall temperature coefficient. A reduction, or the total loss, of the proton beam will then produce an increase of the neutron multiplication coefficient k, which obviously must not bring the reactor critical, not even delayed-critical. 
         [0107]    The effects due to a change of the reactor power are strongly dependent on the actual composition and age of the fuel. They are primarily dependent on two parameters: the thermal conductivity k th  and the thermal capacitance c th  of the fuel elements. Large temperature variations are expected for conventional pin-structured Oxide fuels rods since k th  is relatively low. On the other hand, metal fuel rods have much smaller temperature variations because of high k th . Other fuels, like Carbides or Nitrides are presumably intermediate values between the case of Oxide and the one of metal. 
         [0108]    At each (sudden) change of the fission power, a variation of the fuel temperature is occurring due to the progressive change of the heat stored by c th  and its dissipation to the remainder of the structures through k th . The change in temperature in turn is affecting the value of the neutron multiplication coefficient k. It is noted that k th  will generally decrease very substantially during the natural evolution of the fuel, since it depends on its structural properties, deteriorating with increasing burn-up. 
         [0109]    For illustration purposes, we have considered a large, Lead-cooled subcritical Energy Amplifier of 1.6 GWatt th  and about 50 tons of Thorium Uranium MOX fuel, in the form of standard fuel pins. The Doppler effect, averaged over the whole core, is found to be small and negative,          Δk T           ≈−0.8×10 −5 K −1 . The main temperature effect is due to the fast change in the temperature of the fuel rods, the coolant and the rest of the core having a much smaller effect and generally a much longer time constant. Its time response for a sudden current variation is easily calculated with the help a second order differential equation integrated over the fuel rods and the appropriate compositions. It is well represented by an exponential with a time constant τ th  much shorter than the characteristic time of the delayed neutrons. Representative values are τ th =1.38 s for the initial Thorium Uranium MOX fuel and τ th =3.94 s after a 20% mass burn-up, with an increase of a factor 2.8 from 143° C. to 386.9° C. in the maximum temperature variation of the centre of the fuel pins with respect to the temperature of the coolant. In conclusion, the temperature time response to the neutron multiplication coefficient k is a quantity which should be experimentally measured and periodically monitored during the operation of the sub-critical reactor. 
         [0110]    Having in mind the above-mentioned effects (A), (B) and (C) due to a stepwise change of the proton current, several alternatives are next considered. 
         [0111]    In  FIG. 7 , the event of an inevitable, although rare, beam “trip”, namely a step change bringing the whole proton current promptly to zero at time t=0, is illustrated. The average neutron counting rate R=dC/dt is decaying from an initial value R 0 +RB shown at 25 for t&lt;0 to the semi-stable plateau  26  at t≈0 due to the delayed neutrons, following the curve  20  of  FIG. 5  and exemplified in our case by a value of k which is set to be 1.5 $ away from criticality (level  24  in  FIG. 5 ). 
         [0112]    The reactor temperature will then spontaneously decay, in absence of interventions, causing variations of the neutron counting rate for instance along one of the families of curves as shown in  FIG. 7 . The value chosen is τ th =4 s, corresponding to the worst case of a 20% mass burn-up for the above-exemplified Thorium Uranium MOX fuel. The various curves  27  through  37  represent a fuel-averaged peak fuel core temperature change ΔT max  of 0° C., 100° C., 200° C., 300° C., 400° C., 500° C., 600° C., 700° C., 800° C., 900° C. and 1000° C., respectively. One can see that while for small ΔT max  (27), the counting rate is following the one of the delayed neutrons, as soon as ΔT max  becomes significant, the neutron counting rate is strongly influenced by the changes of k. The estimated value for the previous example after a 20% mass burn-up is near curve  31 . The recorded neutron rate remains stable well above the estimated initial ΔT max . With increasing ΔT max , the neutron counting rate, here due exclusively to the delayed neutrons is extending to longer times, approaching a near constant value when approaching the criticality value, which however is excluded since the delayed neutrons alone will be able to maintain a high temperature in the fuel core. 
         [0113]    In the insert  38  of  FIG. 7 , we show in more detail the first 5 seconds after the “trip”. With adequate statistics, it is possible to smoothly extrapolate with remarkable accuracy the value R 1 +RB of the semi-stable plateau  26  at t≈0. The value of the delayed neutron multiplication k is then extracted with the help of  FIG. 5 . 
         [0114]    The reactivity in $ can also be determined from (1): 
         [0000]    
       
         
           
             
               
                 
                   
                     ρ 
                     
                       β 
                       eff 
                     
                   
                   = 
                   
                     
                       1 
                       - 
                       X 
                     
                     
                       1 
                       - 
                       X 
                       - 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           
                             i 
                             beam 
                           
                           / 
                           
                             i 
                             beam 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             X 
             = 
             
               
                 
                   R 
                    
                   
                       
                   
                    
                   1 
                 
                 + 
                 RB 
               
               
                 
                   R 
                    
                   
                       
                   
                    
                   0 
                 
                 + 
                 RB 
               
             
           
         
       
     
         [0000]    is the ratio of the level R 1 +RB of the semi-stable plateau  26  at t≈0 to the level R 0 +RB of the neutron counting rate  25  at t&lt;0, as indicated in  FIG. 7  (in the case of  FIG. 7 , we have Δi beam =i beam , so RB=0). Clearly, rather than computing ρ/β eff  for a given current drop Δi beam /i beam  and comparing it to a target value or range, it is possible to just compute the related ratio X from the output of the neutron counters and to express the target value or range in terms of X-value. R 0 +RB is directly measured as the stable counting rate prior to the step change of the beam current. Since this rate is stable, there is ample time to get sufficient statistics to measure it reliably. The value R 1 +RB represents the actual counting rate for only a very short period of time, of the order of a few tens of milliseconds, as can be seen in the insert  38 . In practice, the counters  110  may not accumulate enough neutron detection events to provide a reliable measurement in such a short period. However, we can exploit the measured neutron counting rate for a relatively longer time period, more than 100 milliseconds, or even more than 1 second, after the step change of the beam current to obtain a reliable value of R 1 +RB. This is done by extrapolating the values of the neutron counting rate measured after the step change towards t=0, while the beam current is kept off. Extrapolation can be performed using a variety of well-known numerical methods including least mean squares, curve fitting, etc. At t=0, the extrapolated value gives R 1 +RB with a very good accuracy. If needed, the counting statistics can be acquired over several seconds. 
         [0115]    Note that curve  27  represents the situation for a negligible temperature effect (ΔT max ≈0° C.) and curve  37  an averaged centre core temperature ΔT max =1000° C. with respect to the temperature of the coolant. Whatever the temperature scenario, the level R 1 +RB of the semi-stable plateau  26  at t≈0 is safely estimated. 
         [0116]    In reality, the time dependence of the fission rate after a “trip” may have a dependence which is more complex that the one of the simple exponential analysis here illustrated and that in particular the value of τ th  may be different from these elementary expectations. Notwithstanding, the value R 1 +RB at point  26  can be accurately estimated by analytic “continuity” extrapolation along the procedure indicated in the insert  38 . 
         [0117]    A “trip” event having consequences as depicted in  FIG. 7  in terms of neutron population is detected using the current transformers  4 ,  5 ,  10  of the accelerator complex. A few seconds after detection, it is automatically aborted by the prompt insertion of the fast moving “scram” absorber elements  112  with the corresponding large reactivity reduction bringing the fission power down to near zero. However, the present analysis shows that even a failure of the scram system is not causing irreparable damage. Also, before the scram system is activated, it is possible to obtain an estimation of the reactivity ρ (in $) immediately before the beam trip using the estimation procedure described above with reference to  FIG. 7 . 
         [0118]    Other phases of estimating reactivity in the core  100  are used during the normal life of the ADS, in order to monitor the reactivity, or the neutron multiplication coefficient, to make sure that it is in the required range and take any corrective measures using the control rod system  111 . 
         [0119]    Preferably, such phases do not include a complete shutdown of the beam current which, if repeated, may represent a risk for the thermo-mechanical stability of the core. Referring to  FIG. 5 , it can be determined that a step reduction of the beam current i beam  by less than 50% is suitable. 
         [0120]      FIG. 8  is a diagram similar to the one of  FIG. 7  in an example where Δi beam /i beam =0.3. Again, the behavior of (A) the fast component, (B) the delayed component and (C) the temperature variations as a function of time has been taken into account. The neutron counting rate was simulated as a function of time following the step change of the beam current by −30%. After a few seconds, the current was returned back to its original value i beam . An initial k-value corresponding to 1.5 $ below delayed criticality and τ th =4 s were chosen as further parameters. The neutron signal R 0 +RB at 39 is reduced to R 1 +RB at 40, maintaining the full initial contribution due to the delayed neutrons. The various curves  41  through  49  represent a fuel averaged peak fuel core temperature change ΔT max  of 0° C., 40° C., 80° C., 120° C., 160° C., 200° C., 240° C., 320° C. and 400° C., respectively. As expected, as soon as ΔT max  becomes significant, the neutron counting rate is strongly influenced by the changes of k. The estimated value for the previous example after a 20% mass burn-up is near the curve  43 . In the insert  50  of  FIG. 8 , the first 5 seconds following the step change are shown in more detail. Again, it is seen that with an adequate statistics, it is possible to smoothly extrapolate with remarkable accuracy the value R 1 +RB of the semi-stable plateau  40  at t≈0. The value of the neutron multiplication k is then extracted with the help of  FIG. 5  and curve  22 . The reactivity ρ/β eff  can also be estimated using (2). 
         [0121]    The above-described procedure of progressive changes of the accelerator current can be extended during the whole operation of the sub-critical reactor both with negative or positive Δi beam  as required. At each step, the neutron counting rate dC/dt and the corresponding fission-produced power are continuously recorded as a function of time and the new value of the multiplication coefficient k, or dollar value, is calculated. Since the temperature of the fuel is rising with the produced power, the k-value is changing significantly. At each step, control rods are to be moved in order to maintain the required value of k throughout the process. 
         [0122]    Some organized changes of the reactor performance must occasionally take place, including a start-up or shut-down process or a process of varying the reactor power for any reason. The accelerator current is then progressively brought to a required value in a series of several successive increment or decrement steps. Following such step changes of the beam current, the resulting neutron counting rate is accurately measured with a procedure analogue to that described above with reference to  FIG. 7  or  8 : the value for t&gt;0 is extrapolated smoothly towards t=0 from the right side of the curve to extract the value of the semi-stable plateau related to the prompt and delayed neutron components, removing the progressively growing effects of temperature variations. From this extrapolated value divided by the corresponding value for t≦0, one can calculate k in units of $. At each step, the control rods are progressively adjusted, in order to maintain as required the conditions of the nuclear power production setup. The procedure may be optionally repeated in order to optimize the required performance of the reactor. 
         [0123]    It will be appreciated that the embodiments described above is an illustration of the invention disclosed herein and that various modifications can be made without departing from the scope as defined in the appended claims.